Integration and Differentiation in Banach Spaces

Appendix B

Bring me my bow of burning gold! Bring me my arrows of desire! Bring me my spear! O clouds unfold! Bring me my chariot of ﬁre! — William Blake (1757–1827)

In this appendix, we arm us with magic weapons to ﬁght evil integral equations

in ﬁnal frontiers of unexplored Banach spaces. We treat (Bochner) integration,

p ¡ ∞ ¥ ¦¨§ differentiation, function spaces and L p ¢¤£ 1 and similar concepts for functions with values in Banach spaces. We extend standard and extended results on scalar-valued functions for vector-valued ones. Lebesgue measurability, integration and Lp spaces in the scalar-valued case are treated in Section B.1. In the rest of this appendix we treat the extensions of these concepts to the vector-valued case.

In Section B.2, we treat Bochner measurable functions f : Q © B, where B is a Banach space and Q is a positive measure space. In Section B.3, we define and study the Lp and spaces of such functions. In Section B.4, a generalization of the Lebesgue integral (the Bochner integral) is defined and studied for such functions. The reader might wish to have just a look at the beginnings of the sections mentioned above and skip the rest of this appendix until suggested to look up a specific fact by a proof in the main part of this monograph.

Differentiation of integrals and Lebesgue points are treated in Section B.5,

¡ ¡¡

Ω § Ω § §

vector-valued distributions ( ;B : ;B ) are treated in Section B.6,

¡ ¡

k p k p

Ω § Ω § and Sobolev spaces (W ;B and W0 ;B ) in Section B.7.

Throughout this appendix, B, B2 and B3 denote Banach spaces with scalar ﬁeld

K (K C or K R), U, H, and Y denote Hilbert spaces, µ is a complete positive σ measure on a set Q, and is the corresponding -algebra.

We use the (standard) terminology of [Rud86]: a positive measure on a set Q

¡ ¡

is a function µ : 0 ∞ (or the pair µ or the triple Q µ ) s.t.

¢ is a σ-algebra on Q (i.e., is a collection of subsets of Q s.t. Q and is closed under complements and countable unions), and µ is countably additive,

907 908APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

∞

¡ ¡ ¡

¡ ∞ ∞ 0/ § § §

i.e., µ E § ∑ µ E whenever E is disjoint, and µ 0.

k 0 k k 0 k k k 0 (In this chapter, we only treat scalar-valued measures. See Lemma D.1.12 and

Section 2.6 for vector-valued measures.)

¢ §"! σ ∞

We call µ (or Q) -ﬁnite if Q k NQk, where Qk and µ Qk for all

k ¢ N. We call µ complete if all subsets of null sets are measurable.

¥ §

When we assume Q R (or Q ∂D, where we identify ∂D with £ 0 2π ) and

omit µ, we tacitly assume that µ m, the Lebesgue measure (see Theorem 2.20 of

[Rud86]).

A null set is a measurable set N with µ N § 0. A property (e.g. f g for ©

functions f ¥ g : Q B) is said to hold almost everywhere (a.e.) on Q if it holds on ¡

c ¡

# §$ § N : Q N for some null set N. Analogously, we can say that f q g q holds

for almost every (a.e.) q ¢ Q.

¡ § Even though we have allowed a general Q ¥ µ for completeness, we shall

need the results only for 1. the counting measure µ, for which every A Q is

¡ ¡

¥ § measurable and µ A § is the cardinality of elements in A, and for 2. J µ , where

2ωt

J R is an interval (i.e., a connected subset of R) and µ m or dµ e dm

for some ω ¢ R, and even so most results given below are used only for some technical details. Note that these both are complete positive measures, and the latter measure is σ-finite (so is the former too for countable Q). For readers interested in the control of the systems with finite-dimensional in- put and output spaces only (this is very restrictive in cases where the output equals the state), it suffices to consider the (componentwise) Lebesgue measurability and

integral; the Bochner measurability and integral are just the inﬁnite-dimensional *+',* counterparts with &(')& B in place of . We remark that almost all results given for Banach spaces in this appendix are valid for Fréchet spaces, mutatis mutandis (letting B to be an arbitrary Fréchet

p ¡ space would make L Q;B § spaces Fréchet spaces). B.1. THE LEBESGUE INTEGRAL AND LP R; 0 ∞ SPACES 909

p - ∞

/10 243 B.1 The Lebesgue integral and L R; . 0 spaces

The subspace W inherits the other 8 properties of V. And there aren’t even any property taxes. — J. MacKay, Mathematics 134b

Here we shortly recall the Lebesgue integral and Lp spaces from [Rud86].

£65 ¥ ¦

Let R C, or let R be a connected subset of ∞ ∞ (open subsets of

¡ ¡

¥ ¦ ¥ § £75 ¥ § ¥ ¦

£75 ∞ ∞ are arbitrary unions of sets of form a b , ∞ b , a ∞ with

¢8£65 ¥ ¦ a ¥ b ∞ ∞ ; note that R inherits is usual (metric) topology as a subset of

∞ ∞ %

¥ ¦ © £ ¦ £75 ). A function f : Q R is called (Lebesgue) measurable iff f G is

measurable for each open G R. (By Lemma B.2.5(b3), Lebesgue measurability

is a special case of Bochner measurability for R K.)

If f ¥ g : Q R are Lebesgue measurable, then so is max f g , by Theorem

: ¥ §

1.14(b) of [Rud86]; in particular, so are f 9 , where f : max f 0 , f :

¡ §

max 0 ¥;5 f .

¡ =

If µ ¥<§ is any positive measure on Q, and is the collection of sets E Q

¡ ¡ ¡

># §? ¥ @¢A § §

s.t. A E A and µ A A 0 for some A A and we set µ E : µ A

for such E, then the completion µ B¥= 7§ of µ is a (well-deﬁned) complete positive ¡ measure on Q, by Theorem 1.36 of [Rud86]. Note also that rµ ¥<§ is also a

positive measure on Q for any r ¢ R . The Borel (measurable) sets of a: topological space Q are the members of the

minimal σ-algebra containing the open sets of Q. A function f : Q © K is a Borel

£ ¦

(measurable) function if f % V is Borel measurable for all open V K. A Borel

¡ measure on Q is a measure µ ¥<§ s.t. contains the Borel sets (equivalently,

s.t. contains the open sets). The Lebesgue measure m is the completion of a

measure whose domain is the collection of Borel sets, hence a Borel measure.

£75 ¥ ¦ C ¢D£65 ¥ ¦FE G H For f : Q © ∞ ∞ we set esssup f : inf r ∞ ∞ f r a.e. ,

∞ ∞ E ∞ ∞

5 J G !K 'L essinf f : I5 esssup f . We also set r : 0 (0 r ), r : ,

∞ ∞ ∞

M ! !8 ';

r J 0 : (0 r ) and 0 : 0. The function ¢ 1 ¥ q E;

χ ¡ ON

E q § : (B.1)

¢ P 0 ¥ q E

is called the characteristic function of the set E.

¥;QRQ;QR¥ ¥;Q;QRQR¥ ¢M£ ¥ ¦ If n ¢ N, E0 En are disjoint and measurable, and α0 αn 0 ∞

n χ or ¢ B, then s : ∑ α is a simple measurable function and the Bochner k 0 k Ek

integral of such s is given by S n

α ¡ §>Q sdµ : ∑ kµ Ek (B.2) Q

k 0

£ ¥ ∞ ¦ T T

For general measurable f : X © 0 we set Q f dµ : sup Q sdµ, the supre- G mum being taken over all simple measurable functions s s.t. 0 G s f . This integral is usually called the Lebesgue integral, and the term Bochner integral is reserved for functions whose values are not scalar (see Deﬁnition B.4.1).

T ¤T

If Q Q is measurable, then we set f dµ : f dµ.

U U Q Q Q

910APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

! ∞ © £ ¥ ∞ ¦ Let 1 G p . A measurable function f : Q 0 belongs to

1 Y p ¡

p ¡ p ∞

¥ ∞ ¦¨§ & & WV X ! ∞ £ ¥ ∞ ¦B§ & &

L Q; £ 0 if f p : Q f dµ , and to L Q; 0 if f ∞ :

*4! ∞ & & & & Z & &

* p p

\^] esssupQ f (we sometimes write f L or f L Q; [ 0 ∞ instead of f p).

p :

G ∞ To be exact, the L spaces (1 G p ) are quotient spaces over with the

p ¦ set of functions that are 0 a.e. (i.e., L is the space of equivalence classes £ f ,

p ¡

¦ £ ¥ ¦¨§ where g ¢£ f iff g f a.e.). Thus, L Q; 0 ∞ becomes “a normed space with

p ¡

¥ ¦ £ ¥ ¦B§

scalar ﬁeld £ 0 ∞ ”. (One easily veriﬁes that L Q; 0 ∞ is a vector space with

¥ ¦ scalar field £ 0 ∞ and that the axioms of a normed space are satisfied w.r.t. this scalar field; the latter requires the Minkovski Inequality, Theorem 3.19 of [Rud86],

∞

& G_& & ¤& & ¥ © £ ¥ ¦ which says that & f g p f p g p for all measurable f g : Q 0 .)

∞

& G`& & & & ¥ ¢a£ ¥ ¦

We also note the Hölder Inequality: & f g 1 f p g q for p q 1 s.t.

J 1 J p 1 q 1. The reader will later note that these two inequalities extend to

vector-valued functions, because for measurable (see Deﬁnition B.2.1) f : Q © B

& <&b& & &

we shall deﬁne & f p : f B p. H

Lebesgue’s Monotone Convergence Theorem says that if C fn are measurable,

¡ ¡

G Gc'R';'dG ∞ §e© § ¢

0 G f1 f2 a.e. on Q and fn q f q for a.e. q Q, then f is ©fT measurable and T Q fn dµ Q fn dµ.

n H

Lemma B.1.1 Let Q R be measurable. For any measurable sets C E s.t.

¢ C H m E §hg 0 for all n N, there are disjoint measurable E s.t. E E and

n n n n

∞ ¡ §?g ¢ g m En 0 for each n N.

∞ ¡

§ig ¢

Proof: Choose F E s.t. g m F 0. For each n N 1, choose

0 0 0

¡ n ∞

! §! % § # kjkjljk

F E s.t. 0 m F 2 max m F and set E : F F .

n n n k 0 n 1 k n n k n 1 k

¡ ¡ ¡

¡ ∞ k

§h! §mG § §mg Then m F ∑ 2 % m F m F , hence m E 0, for any n n

k n 1 k k n 1 n

: n n ¢ N.

(See the notes on p. 947.)

B.2. BOCHNER MEASURABILITY (F L Q;B ) 911

- 3 B.2 Bochner measurability ( f o L Q;B )

You cannot have a science without measurement. — R. W. Hamming

In this section we treat Bochner measurability in order to be able to define Lp spaces and the Bochner integral in the next two sections. For most readers, it suffices to just have a look at Definition B.2.1 and Lemma B.2.5 so as to be convinced that Bochner measurability is analogous to Lebesgue measurability.

A function s : Q © B is called a countably-valued measurable function iff it

χ ¢

∑ can be written as s x , where the sets E Q are measurable and x B k N k Ek k k

(k ¢ N). In general, Bochner measurability is deﬁned as follows:

Deﬁnition B.2.1 (Bochner measurability) A function f : Q © B is (Bochner)

1 ¡ § measurable , denoted by f ¢ L Q;B , if some sequence of countably-valued measurable functions converges to f a.e.

A function f : Q © B is called almost separably-valued if it can be redeﬁned ¦

on a null set (a set of measure zero) so that f £ Q becomes separable. © In Lemma B.2.5(g) we shall show that a function f : Q © K (or f : Q

∞ ∞

¥ ¦ £75 ) is measurable iff it is Lebesgue measurable. We follow the standard convention to identify functions equal a.e. as members

of L. Thus, the elements of L are actually equivalence classes. We sometimes ¦ write “ £ f ” instead of “ f ”, when it would otherwise not be obvious that we refer

to the class, not to the function.

¡ ¡

§ ¢ § p

Let f ¢ L Q;B and F B;B3 . Then F f is measurable (because

F p sn F f a.e.). If f g a.e., then F f F g a.e. Thus, we can follow the

¦ r£ p ¦

standard convention to deﬁne F pq£ f : F f . Most common examples of this

¦ _£ ¦ £ ¦st£ ¦ `£ ¦ ¥ ¢ § ¢ are the deﬁnitions α £ f : α f and f g : f g for f g L Q;B , α K. We also extend to classes any other operations that can be well deﬁned through representatives.

For separability results we shall need the following:

§ ¢

Lemma B.2.2 Let the sets Ak be at most countable k ¢ N , and let n N. Then

E E

';';' C H C

u u u

the sets A A A , A , B A B has n elements and B A B

0 1 n k N k 0 0

E E is ﬁnite H are at most countable. A A

For any set A, cardA ! card2 (recall that 2 is the set of all subsets of

A). If A and B are nonempty sets, (at least) one of which is inﬁnite, then

¡ ¡

§$ C ¥ Hi § u card A B max cardA cardB card A B . n (The countability claims are easily deduced from Theorems 2.12 and 2.13 of [Rud76]; the general cardinality claims are also well known and follow from, e.g., Theorems 176 and 180, pp. 276 and 280 of [Kelley].) Next we list a few separability results in order to be able to prove Lemma B.2.5.

Lemma B.2.3 (Separability) Let E B be separable for all n ¢ N. Then n 1The term “strongly measurable” is sometimes used, but we reserve it for Deﬁnition F.1.1. 912APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

(a) The closure and span of E0 are separable.

(b) If E E , then E is separable. 0

(d) If E B is weakly separable, then it is separable.

(e) Subsets of Rn are separable.

¥ § £ ¦ (e) If Q is separable and f ¢ Q Q , then f Q is separable.

Proof: Let S E be dense in E and countable, for each n. n n n

(a) The set S0 is dense in E¯0 and the set of ﬁnite linear combinations of

¥ ¢ elements of S0 with coefﬁcients of form r iq (q r Q) is dense in the span of E0.

n n

cC H ¥ ¢ ¢

(b) Let S0 xk k N. For each n k N, choose an element xk Ek :

C ¢ 5 &v! J §

y E & xk y 1 n 1 , if such an element exists. Then the union

E g S of these elements is dense in E, because if x ¢ E and N 0, we can

5 *w! J ! choose xk s.t. * x xk 1 2N, so that Ek is nonempty for n 2N, hence

2N 2N

5 &xG_& 5 &y¤& 5 &x! J

& x xk x xk xk xk 1 N. Thus, S is dense in E. (c) The union nSn is dense in nEn.

(d) Let S E be countable and dense in the weak topology of B. Let M be the closure of the span of E. Then M is weakly separable, by (a) (whose proof is valid for any topological vector space), hence M is separable, by Theorem 3.12 of [Rud73], hence E is separable, by (b).

(e) The countable set Qn is dense in Rn, hence (e) follows from (b).

¦ n

(f) If S Q is dense, then f £ S Q is obviously dense.

Ω ¡ Ω

¥ §;§ Lemma B.2.4 Let B be separable and C. If f ¢ ; B B , then

¡ Ω ¥ §;§ f ¢ ; B B for some separable closed subspace B B . 2 2 2 Ω Ω

Proof: Let and Q B be dense and countable. Then

Ω Ω £ z¦£ ¦ £ ¦£ ¦ © © B k : f Q f B is dense and countable, by continuity (x x & s

2 k k

¡ ¡

§ © § k

s { f sk xk f s x). Consequently, the closed span B2 of B2 is a separable

¡ ¡ ¡

Ω ¦£ ¦ §|£ ¦ §"¢} ¥ § Banach space, and it contains f £ B , i.e., f s B B (i.e., f s B B )

2 2 n for any s ¢ Ω.

Now we are ready to list the standard properties of Bochner measurability:

¥ © ¢

Lemma B.2.5 (Bochner Measurability) Let fn ¥ g h : Q B be measurable (n β ¢ N) and α ¥ K.

(a1) The function αg βh is measurable.

¡ ¡

¥ § ¢ ¥ §

(a2) If T ¢~ B B2 (or T B B2 ), then Tg is measurable.

¥ §

(a3) If f : Q © B B2 is measurable, then so is f .

(a4) If also f : Q B2 is measurable, then so is f g : Q B u B2.

¥ § © ©

Thus, then B f g : Q B3 is measurable for any continuous B : B u B2 B3.

B.2. BOCHNER MEASURABILITY (F L Q;B ) 913

Λ Λ ¢ (b1) A function f : Q © B is measurable iff f is measurable for all B (or Λ for all in a norming subset of B ) and f is almost separably-valued.

(b2) A function f : Q © B is measurable iff some sequence of countably-valued

measurable functions converges to f uniformly outside some null set.

{ { (b3) Let f : Q © B. Then (i) (ii) (iii); if B is separable, then (i)–(iii) are equivalent:

(i) f is measurable;

1 ¡

£ ¦ C ¢ §h¢ H (ii) the set f % V : q Q f q V is measurable for each open

V B; E

(iii) Λ f is measurable for all Λ ¢ B .

∞ ¥ ∞ ¦

This holds also with £75 in place of B.

¡ ¡

¢ §© § ¢ (c) If fn : Q © B (n N) are measurable and fn t f t for a.e. t Q, then

f is measurable.

¢ ©

(d1) Let Q Q be measurable for all n N and Q Q . Then f : Q B

n n N n ¢ is measurable iff f is measurable Q © Q for all n N.

* n Qn

(d2) If B is a closed subspace of B and f Q B , then f is measurable Q © B 2 2 2

iff f is measurable Q © B.

u u

(d3) A function f : Q B1 u B2 Bk is measurable iff f j is measurable ¥RQ;Q;QR¥ for j 1 k.

(e) Any continuous function Q © B is measurable if Q is separable and µ is a n

Borel measure (e.g., µ m and Q R is measurable).

¡ ¡

§ § 9 (f) If B K, then Reg 9 , Img are measurable.

In (a4) the map B can be multiplication, inner product, a continuous bilinear

∞ & © £ ¥ § map, or similar. By (a2), & g B : Q 0 is measurable.

Note for (d1) that the restriction of µ to some measurable Q Q is also a

complete positive measure. Note also that condition (ii) in (b3) depends on (and B) only, whereas conditions (i) and (iii) depend on the measure too.

φ φ 1 ¢ See Lemma B.4.10 for the measurability of f p with increasing. By (d1), piecewise continuous functions are Borel-measurable (hence

Lebesgue-measurable if Q Rn with Rn’s topology).

in the deﬁnition of measurability. Then αgn βhn is countably-valued and measurable for all n ¢ N and converges to αg βh a.e. The proofs of (a2), (a3),

(d1), (d2) and (d3) are more or less analogous.

E σ

¤C ¢v H

Note for (d1) that U : E E Q is a -algebra on Q , and µ

Q E Q U

is a complete, positive measure on µQ U .

(Claim (a3) is a special case of (a2), it holds to both Hilbert space adjoint

©f ¥ §

f : Q B2 B (in case that B and B2 are Hilbert spaces) and to the Banach

operator). Claim (a2) holds even when T is only continuous from B to the weak topology of B2 (use (b1) and Lemma B.2.3(d)).) (b1) Without our reference to norming sets, the proof on pp. 72–73 of [HP] applies. Assume then that f is almost separably-valued and Λ f is measurable

for all ¢ C, where C B is a norming set.

Redeﬁne f on a null set so that f £ Q is separable, and replace B by the closed ¦

span of f £ Q , which is separable. Let S B be dense, and choose countable

& * * ¢ & & * *

C C s.t. & x sup Λx for all x S Then x sup Λx for all

U U Λ Λ

B C B C

¡ ¡

& §)& * §)* ¢ & &

x ¢ B. It follows that f t sup Λ f t for all t Q, therefore, f Λ

B C U B

5 & ¢ is measurable; analogously, so is & f x B for any x B. Thus, the rest of the proof on pp. 72–73 of [HP] is valid. (b2) This is Corollary 1 on p. 73 of [HP]. (b3) The second claim follows from the ﬁrst and (b1), so we only need to

prove the ﬁrst claim.

r£75 ∞ ¥ ∞ ¦ 1 (i) (ii) (iii) for B K and for B : This can be deduced

from Theorems 1.14 and 1.17 of [Rud86] (note that (i) (iii) is trivial, because

K " K).

{ Λ ¢ Λ p §|£ ¦ 2 (ii) (iii): If f satisﬁes (ii), B , and V K is open, then f V

1 1 1

£ % £ ¦¦ % £ ¦ f % Λ V is measurable, because Λ V is open. Thus, Λ f is measurable,

by 1 .

{ C H 3 (i) (ii): Assume (i), so that some sequence sn of countably-valued,

measurable functions converge to f everywhere (redeﬁne sn and f to be zero

¡ §

on a null set, if necessary). Let V B be open, and deﬁne F ¢ B;K by

¡ c

§ ¥ § & 5 & p © p

F x : d x V : infy Vc x y B. Then F sn F f everywhere, hence

¢ § p ¢ F p f L Q , because F sn is countably-valued and measurable for all n N.

1 1 1 ¡ 1

£ ¦ £ £ #C H4¦¦4 p § £ #C H4¦

% % %

Therefore, the set f % V f F K 0 F f K 0 is open,

{

because F p f satisﬁes (ii), by the implication “(i) (ii)” of 1 . { 4 (iii) (i) for separable B: This follows from (b1).

(Note that I : B © B satisﬁes (ii) but not (i) if B is an unseparable Banach space with the counting measure.) (c) By Theorem 1.14 of [Rud86], Λh is measurable, so we only have to

show that h is almost separably-valued. ¡

¡ c

§ ¢

Choose a null set N Q s.t. h t §?© h t for t N . For each n, choose n

¦ #

a null set Nn s.t. hn £ Nn is separable. Let N : N nNn, Q : Q N . Then

£ z¦

E : nhn Q is separable, hence so is the closed subspace M of B spanned by

¡ ¡

§( §¢ ¢

E. But h t limn ∞ hn t M for each t Q , hence h is almost separably- valued. :

£ ¦ £ ¦ (e) By Lemma B.2.3(e), f Q is separable. Because f % V is open, hence

measurable for each open V B, f is measurable, by (b3).

¡ §

(f) By (a2), Reg and Img are measurable; by Section B.1, so are Reg 9 ,

¡ § Img 9 .

(g) This follows from (b3). n

The rest of this section consists of less important results, hence the reader might wish to skip them. B.2. BOCHNER MEASURABILITY (F L Q;B ) 915

We will use the following lemma to generalize several scalar results to the vector-valued case.

Lemma B.2.6 Let f be Bochner measurable. Then f 0 a.e. iff f 0 a.e. for

Λ Λ all ¢ B (or for all in a norming subset of B ).

Λ Λ

If f is only “weakly vector measurable”, i.e., f is measurable for all ¢ B ,

¢ P

then we may have Λ f 0 a.e. for all Λ B even though f 0 everywhere (set ¡

¡ 2

§ C H §

f t : e , where e is the natural base of B : R (i.e., e : ), so that

t t t R t t

2 ¡

¢ § §>¥ ¢ P C H ∑ α

for any y : k ketk R we have f t y B 0 for t k tk ). &q

Proof of Lemma B.2.6: The necessity is clear, so we assume that & f

§?g ¢ P

ε g 0 on E R and m E 0, and ﬁnd Λ B s.t. Λ f 0 on a set of positive

measure.

£ ¦ C H

W.l.o.g. we assume that f E has a dense countable subset bk k N. For

¡ ¡

E ε

§qg cC ¢ & §$5 &! J H

some k ¢ N, we have m Ak 0, where Ak : r E f r bk 3 . E

Λ Λ Λ ¡ Λ ε *g & &>J & &G * §,*xg* *15 J g

If * bk bk 2 and 1, then f r bk 3 : M

ε Λ n &>J 5 J g ¢ & &

& bk 2 2 0 for r Ak. Thus, f ∞ M 0.

g § C ¢ E & 5 &e! H

For x ¢ B and r 0 we set Dr x : x B x x r .

¡ ¡

§ ©

Lemma B.2.7 (Ess range f § ) Let f : Q B be measurable. Then

¡ ¡

¡ 1

C ¢ g { £ §¦¨§"g H

essrange f § : x B r 0 µ f Dr x 0 (B.3)

¡ ¡

¡ 1 c

£ § ¦¨§ § is closed and µ f % essrange f 0. Moreover, essrange f is the smallest

set with these properties.

¡ § Proof: One easily veriﬁes that E f : essrange f is closed. We assume

c H

w.l.o.g. (see Lemma B.2.5(b1)) that B is separable. Let C xk be dense in E f . ¡

¡ 1 c

C g £ §¦¨§? H For each k, set r : sup r 0 µ f % D x 0 . Obviously, E V :

k r k f

¡ ¡ ¡

¡ 1 1

§ £ ¦B§( £ §¦¨§(

% % Y kDrk Y 2 xk ; but µ f V 0, because µ f Drk 2 xk 0 for all k; hence

¡ 1 c

£ ¦¨§$ µ f % E f 0. By the deﬁnition of E , the set Ec contains any open set G B s.t. f f

¡ 1 c

£ ¦¨§$ n µ f % G 0, i.e., E f is the biggest of such sets.

Lemma B.2.8 Let f : Q © B be measurable.

¡ ¡

¢ g §$g

(a) If µ Q §$g 0, then there is a0 Q s.t. for each ε 0 we have µ Aε 0, where

¡ ¡

¢ & §y5 §)& ! H

Aε : DC a Q f a f a0 B ε .

¡ §g

(b) If(f) f is not zero a.e., then there are A Q and Λ B s.t. µ A 0 and Λ

Re f g 1 on A.

C ¢

If, in addition, B0 is a closed subspace of B and µ E §g 0, where E : t

E Λ Λ

¢ H

Q f t §qJ B0 , then we can choose and A so that 0 on B0. E

1 ¡

¢ £ §¦

Proof: (a) Choose any a0 f % essrange f .

H ¢C ¥ ¥ ¥;QRQ;QzH

(b) If B0 has not been given, set B0 cC 0 . Choose n 1 2 3 s.t.

¡ ¡

¢ §>¥ §g J H ¢

An : DC q Q d f q B0 1 n has a positive measure. Choose then a0 An E

916APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES ¡

ε E J `C ¢ 5 §)& ! J H for f and : 1 2n as in (a). Then V : b B & b f a 1 2n is * A 0 B

n E

/ 1

£ ¦ 0

convex, open and nonempty, V B0 , A0 : f % V has a positive measure

(by (a), because A0 A0 An), and also B0 is convex and nonempty, hence

there are Λ ¢ B and γ R s.t.

γ G Λ ¢ ¥ ¢ ¥ ReΛx ! Re y for all x V y B0 (B.4)

Λ ¦

by Theorem 3.4 of [Rud73]. Thus, £ B0 is a proper subspace of K, hence

¡ E

Λ Λ ¦ C H ¢AC ¥ ¥ ¥RQ;Q;QzH C ¢ §?!K5 J H

£ B0 0 . Find k 1 2 3 s.t. A : a A0 Re f a 1 k has a

E Λ Λ n positive measure, and then replace by 5 k .

The following result can be used for convolutions: ¡

n ¡

¥ §© 5 § Lemma B.2.9 Let f : R © B be measurable. Then r s f r s is measur-

n n ©

able R u R B.

¡ ¡

¡ ∞ n § © ¢ # §? Proof: Let sn r §?© f r , as n , for all r R N, where m N 0

and sn is countably-valued and measurable (n ¢ N) (see Deﬁnition B.2.1). We ∞ nχ n may and do require that s ∑ x n where E is Borel measurable for each

n k 0 k Ek k

k ¥ n (by redeﬁning each sn on a null set).

S S S S

¥ § 5 ¢ H

Set N2 : ¤C r s r s N . Then (see Theorems 8.11 and 8.12 of [Rud86])

¡ ¡

¡ χ χ

¥ § § Q m N2 § N2 r s dr ds s N r dr ds 0 (B.5)

Rn Rn Rn Rn :

¡ ¡ ¡ ¡

¡ c

§"© 5 § ¥ § ¢ ¥ §© 5 §

But sn r 5 s f r s for all r s N2, and r s sn r s is countably-

§(© 5 n valued and Borel measurable, because r¥ s r s is a Borel function.

(See the notes on p. 947.) B.3. LEBESGUE SPACES (LP Q µ;B ) 917

p - 3 B.3 Lebesgue spaces (L Q / µ;B )

If God is perfect, why did He create discontinuous functions?

In this section, we ﬁrst deﬁne the Lp spaces (“Lebesgue spaces”) and then go on to prove several technical results, some of which are considered to be well- known although not easily found in the literature.

∞ © £ ¥ ¦

The continuity of &'& B : B 0 implies that if f is measurable, then so &

is & f B. Thus we can make the following deﬁnition:

¡ ¡ ¡

p ¡ p p

§ G G § ¥ §

Deﬁnition B.3.1 (L Q;B § ) Let 1 p ∞. L Q;B : L Q µ;B is the space

& &

of (equivalence classes of) measurable functions f : Q © B whose norm f p :

¡ ¡

¡ p p

'z§)& & Z ¥ § ¥ § && p f B L Q ] is ﬁnite. We set L Q µ : L Q µ;K .

p ¡

¥ ¥ § If B is a Hilbert space, then we set f g L2 : Q f g B dµ. By Q;B we

p ¡ mean L Q;B § with the counting measure.

If compact subsets of Q are measurable, then we set ¡

p ¡

§ C ¢ § & & ! HFQ ¥ ∞

p Z

] L Q µ;B : f L Q;B f for all compact K Q (B.6)

loc L K µ;B E σ p

If, in addition, Q is -compact, then we equip Lloc with the topology induced by

& Z

& p ] the seminorms f L K µ;B (see Theorem 1.37 of [Rud73]); in particular, then

p p

p Z

] f f in L if f f 0 for all compact K Q (and f f L n loc n L K µ;B n loc

for all n).

& £ ¦d £ ¦ &?'& Note that & f p 0 iff f 0 a.e., i.e., if f 0 . Thus, p becomes a norm on Lp (with equivalence classes as elements). We deﬁne the “quasinorms”

1 Y p ¡

p ¡ p

& WV & & X ¢ ¥ § ¢ ¥ § & f p : Q f B dµ for p 0 1 too (but the vector spaces L , p 0 1 p are not normed spaces, cf. [Rud73]). The topology of Lloc is only rarely needed, hence the reader may well skip it; the resulting convergence condition given at the

end of the above deﬁnition is only slightly more useful.

¡ ¡ ¡

p ¡ p

§ ¢£ ¥ ∞ ¦B§ §! ∞ Obviously, L Q;B § L Q;B p 1 . If µ K for all compact

loc ¡

p ¡ r

§ subsets of Q, then L Q;B § L Q;B (∞ p r 1), by the Hölder loc loc

inequality; if, in addition, Q or B is separable and µ is a Borel-measure, then

¡ ¡ ¡

¡ p ∞

§ § § Q;B § L Q;B . One usually equips Q;B with L Q;B topology, but loc loc we do not need this.

p ¡

Theorem B.3.2 The space L Q;B § is a Banach space (a Hilbert space if p 2

C H and B is a Hilbert space). If fn © f in L , then some subsequence of fn converges to f a.e.

Ω n Ω p ¡ Ω § If R is open and µ is a Borel measure on , then L ¥ µ;B is a Fréchet loc space (hence a complete metric TVS).

p q ∞

Note that if fn © f in L , fn g in L and fn h a.e. pointwise, as n ,

C H then f g h a.e. (take a subsequence of fn converging pointwise a.e. to f and g).

Proof: 1 The proof of the ﬁrst paragraph is identical to the scalar case (e.g., [Rud86, Theorems 3.11 & 3.12]), and hence omitted. 918APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

C H 2 Claims on Lloc: Let Kk k N be as in Lemma A.2.3. One easily veriﬁes

p ¡

Z Ω § &"'& p that the norms L Kk;B ] generate the topology of Lloc ;B ; in particular, p Lloc is metrizable, by Remark 1.38(c) of [Rud73].

p k

H C H Let C fn be a Cauchy-sequence in Lloc. Then fn has a limit f in each

p ¡ χ k k

L Kk ¥ µ;B . Set f : ∑ f . Obviously, f f a.e. on each Kk and

k Kk Kk 1 ¡

¡ p p

§ © § © ¢ Ω f L ;B , hence fn f in each L Kk;B , so that fn f in Lloc. Because

p p H C fn was arbitrary, Lloc is complete. By Theorem 1.37 of [Rud73], Lloc is a

Fréchet space (i.e., a complete locally convex metrizable TVS). n

Also most other standard analysis results extend to the vector-valued case;

p ¡ ! in particular, standard properties of L K § spaces (e.g., reﬂexivity (when 1

p ¡ n

G ! § p ! ∞) and separability (when 1 p ∞)) are inherited by L R ;B if(f) also B possesses the same property; see [DU]. The most important exceptions

p ¡ (fortunately not much needed in this monograph) are that the dual of L Q;B §

q ¡ need not be L Q;B L§ (see [DU, Theorem IV.1.1, p. 98]; cf. Lemma B.4.15), and that an absolutely continuous measures (and functions) with values in B need not be differentiable, unless B is a Radon–Nikodym space (and µ is σ-ﬁnite and

∞ ! 1 G p ). For most purposes, it suffices to know that Hilbert spaces and other reflexive spaces are Radon–Nikodym spaces [DU, p. 61, p. 98 & p. 82]. Derivatives are defined as in the scalar case: d ff

Deﬁnition B.3.3 ( dtt ) Let J be an interval of R. The derivative of a function ¢

f : J © B at t J is

¡ ¡

§y5 §

f t h f t

f t § : lim (B.7)

h 0 t h J h :

Ω n Ω

¢ ¢C ¥ ¥;QRQ;Q¥ H

Let R be open, n ¢ N 1, q , j 1 2 n . Then the jth partial

¡ ¡ ¡

¡ Ω

§ § © © § derivative D jg § q : g j q of g : B at q is the derivative of h g q he j at 0. If g has all n partial derivatives at q [and they are continuous], then g is [continuously] differentiable at q. If g is [continuously] differentiable at each point of Ω, then g is [continuously] differentiable (on Ω). α

1 αn 'R';' The partial derivatives of g of order k ¢ N are the functions D1 Dn g for

n n α which α ¢ N , ∑ k. If all partial derivatives of g of order k exist [at q], j 1 j

then g is k times differentiable [at q].

¡ ¡

¥ ¥ ¥;Q;Q;Q¥ § ¥ ¥ ¥ ¥;Q;QRQ¥ §

(Here e j is the jth unit vector (e1 : 1 0 0 0 , e2 : 0 1 0 0 0 , ...).)

¡ ¡ ¡ ¡ ¡

§ § §R§ ¢

Obviously, the deﬁnition of f t § implies that T f t T f t for T

¡ ¡

¥ § § B B2 whenever f t exists. Note that we allow one-sided derivatives at the

endpoints of J (if they belong to J).

Let Q be a topological space. The space Q;B § is the vector space of

¡ §

continuous functions f : Ω © B. We equip the following subspaces of Q;B

q Q B b

bounded), bu ( f bounded and uniformly continuous; this requires that Q is

¢ g & &¡! metric), : IC f for any ε 0, there is compact K Q s.t. f ε on

0 b

H OC ¢

Q # K (often called as “the functions vanishing at inﬁnity”), and c : f

supp f is compact H . Obviously, .

c 0 b E B.3. LEBESGUE SPACES (LP Q µ;B ) 919

n ¢

Let Ω ¢ R be open (or an interval). Let be one of the symbols

0 k 1 ¡

¥ ¥ ¥ ¥ ¢ I¢ ¢ Ω § DC ¢£¢ Ω § ¢

b bu 0 c. Then we set : , : ;B : f ;B D j f

¡ ¡

k ¡ ∞ k

§ ¢AC ¥ ¥;QRQ;Q+¥ HbH ¢ ¢ § ¤ ¢ § ¢ Ω Ω Ω

;B for all j 1 2 n (when k N), ;B : k N ;B .

Lemma B.3.4 ( ) Let Ω be a metric space. Then Ω;B §

c 0 bu b

c 0 bu b c

¡ ¡ ¡

§ §

Ω;B § Ω;B Ω;B ; the spaces , and are Banach spaces

0 bu b 0 bu b

(under the supremum norm), and c is dense in 0.

¡ ¡ ¡ ¡

§ § § §

Proof: 1 Claims Ω;B Ω;B Ω;B Ω;B : These

c 0 bu b

¡ Ω §

are obvious except the uniform continuity of function f ¢ 0 ;B . Given

¡ ¡

Ω ε Ω E ε

§ g cC ¢ & §)&¤ J H

f ¢ 0 ;B and any 0, set K : q f q 3 , so that K is

a closed subset of a compact set, hence compact. Analogously, K : cC q

¡ ¡ E

Ω E ε Ω ε §)&i J H IC ¢ §,&qg J H &

& f q 2 is compact. Since K V : q f q 3 K

E E ¡

δ ¡ c c ¥¥ §? ¨¥ §?g

and V is open, we have : d K K d K V 0, by Lemma A.2.1(c).

¡ ¡ ¡

& §15 ¦§)& ! ε ¥ ¢ ¥ 7§(! δ

Choose δ g 0 s.t. f q f q B for all q q K s.t. d q q . Then,

¡ ¡ ¡ ¡

¢ Ω ¥ 7§?! δ ¥ δ ¦§ & §y5 6§)& ! ε

given q ¥ q s.t. d q q min , we have f q f q B .

¡ ¡

¢ yP¢ & §§5 7§)& ! J J !

Indeed, if q P K, then q K , hence then f q f q B ε 3 ε 2 ε;

¢ ¥ ¢ the case with q P K is analogous, and the case q q K follows from the

deﬁnition of δ.

C H

2 The completeness of 0, bu and b: Let fn be a b-Cauchy-sequence.

¡ ¡ ¡

§¨H § §©¢ ¢ C ∞

Then so is fn q , hence f q : limn fn q B exists, for each q Q. ¡

ε ¡ ε

¢ & 5 &«ª & §w5 §)& ! J Given g 0, there is N N s.t. f f : sup f q f q 2

n m b q Q n m B ¡

¡ ε ε & §5 §,& G J ! ¥ ∞ for all n m N, so that sup lim f q f q 2 for all q Q n n m B

∞ © ©

m N. Consequently, fm f uniformly as m . As one easily veriﬁes, it

follows that f is continuous and bounded. hence fn © f in b.

ε g & 5 &! ε J If fn ¢ 0 for all n, then, for each 0 we can choose N s.t. fN f 2

c c

&! ε J & &! ε J ε J ¢

and K s.t. & fN 2 on K , so that f 2 2 on K ; thus, then f 0.

Finally, assume that fn ¢ bu for each n. Given ε 0, choose N s.t.

¡ ¡

5 &q! J g & §y5 7§)&©! J

& fN f ε 3, and choose then δ 0 s.t. fN q fN q ε 3 whenever

¡ ¡ ¡

¢ ¥ 7§¤! & §5 7§)&¬G J J J

q ¥ q Ω, d q q δ. Then f q f q ε 3 ε 3 ε 3 whenever

1¢ ¥ §! ¢

q ¥ q Ω, d q q δ. Thus, then f bu.

3 c is dense in 0: (This holds also when Ω is a locally compact or normal

(possibly non-metrizable) Hausdorff space.)

¡ Ω ε δ § g ¥ B¥ §

Let f ¢ 0 ;B and 0. Choose K K as in 1 . Set g q : 1 in

¡ c c

§ ¢ ¥L£ ¥ ¦¨§

K and g q : 0 in K , so that g K K 0 1 . By Tietze’s Extension

Ω £ ¥ ¦¨§

Theorem [Kelley], g has an extension h ¢ ; 0 1 . But supph K, hence

¡ ¡

Ω § ¢ Ω § & 5 &eG ε J

h ¢ c . It follows that h f c ;B and h f f 2 (since h f f on

¡ c

& 5 &hG<& &h! ε J ¦§ ¢ Ω § ε g

K and h f f f 2 on K ). Since f 0 ;B and 0 were

arbitrary, c is dense in 0. n

The sequence spaces cc (of ﬁnite sequences) and co (of vanishing sequences)

are sometimes needed; we brieﬂy introduce them below as special cases of c and

¡ 1 when q q ;

6§ O®

For any set Q, the function d q ¥ q : is a metric, called the

U 0 when q q discrete metric of Q. In the corresponding discrete topology every subset of Q

is open, hence every function Q © B is continuous. A subset of Q is compact

920APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

¡ ¡ ¡ ¡

§ § § `C ©

iff it is ﬁnite. We set co Q;B § : 0 Q;B and cc Q;B : c Q;B f : Q

¢ H

B f q § 0 except for ﬁnitely many q Q , where Q is equipped by its discrete E

¡ /

§ P

topology. (Recall that empty set is ﬁnite, i.e., 0 ¢ cc Q;B (for Q 0).) Thus,

¡ ¡

¡ ∞

§ § co Q;B § is the closure of cc Q;B under the supremum norm (i.e., in Q;B ,

the set of bounded functions Q © B), by Lemma B.3.4.

The following two lemmas contain important facts:

Lemma B.3.5 (f is measurable) Let J R be an interval and let f : J B be

§ ¢

Lebesgue measurable. If f t exists for a.e. t J, then f is Lebesgue measurable. n

(This follows from Lemma B.2.5(c).) ¡

p ¡

¥ ¦ ¢ § ¢t ¥ § ¢

Lemma B.3.6 Let p ¢M£ 1 ∞ . If f L Q;B and T B B2 , then T f

¡ ¡ ¡ ¡

p ¡ p p

§ ¢ ¥ §;§ ¢ ¥ §;§ L Q;B2 . If F L Q; B B2 , then F L Q; B2 B . n

(This follows easily from Lemma B.2.5(a2)&(a3). This implies that the Bochner integral is a special case of so called Pettis integral.) A casual reader might wish skip the rest this section except Lemma B.3.9 and Theorem B.3.11(a1)&(b1). Most other results below are rather technical and less often needed. p p ∞

Most “L mass” of a L function lies on a compact set (unless p ):

¡ ¡

p ¡ n n

§ ¢ ¥ ∞ § ¢ § ∞ Lemma B.3.7 Let f ¢ L R ;B and p 0 , or f 0 R ;B and p .

& © © ∞ ε ¥ g Then & f 0 as R . In particular, for any r 0, there is

n p

q R q R

° ±

χ E° τs ε n

g & & ! ¢ * *4g

R 0 s.t. f for s R s.t. s R. n

n p

q R q r

E+° ° ²

(This follows from the (scalar) Monotone Convergence Theorem (note that

*4g 5 * *4G * q s R r when q r), except in the (trivial) 0 case.)

t p ¡

π τ ©

Corollary B.3.8 (π τ f © 0) Let f L R;B and 1 p ∞.

: :

τt τT : p π π π

] [ ]

Then f f in L , as t T R, and [ T t f f , 0 t f f ,

: %

π π τ ¡ p ∞

1 f 0 and t f 0 in L , as t T .

]

[ 0 t :

p ¡ π τ t ∞ ¢ § & & © & & ©f %

If g Lloc R;B , then g p g p as t . n :

(One obtains this easily from Lemma B.3.7.)

∞ p ¡ n ¥ § ¢ § Lemma B.3.9 Let p ¢££ 1 and f L R ;B . Then

τ ¡ n

5 § & © © Q

& f h f p 0 as h 0 in R (B.8)

¡ n § Proof: By uniform continuity, (B.8) holds for f ¢ c R ;B . For general

ε g φ ¢ & 5 φ & ! ε J δ g

f ¢ L and 0, choose c s.t. f p 3, and then choose 0 s.t.

¡ ¡

φ 5 τ § φ & ! ε J * *,! δ & 5 τ § & ! ε J ε J ε J ε

& h p 3 for h . Then f h f p 3 3 3 for

*4! n * h δ.

Characteristic functions can be approximated by smooth functions:

B.3. LEBESGUE SPACES (LP Q µ;B ) 921

n ∞ ¡ Ω §

Lemma B.3.10 If K Ω R , K is compact, and Ω is open, there is φ ¢ c

χ χ φ G s.t. K G Ω.

n n ¡

Ω Ω ∞ ∞ ¢M£ ¥ §

If R is open, E R , m E §¡! , p 1 , [and w 0 is

1 ε Ω L on a neighborhood of E], and g 0, then we can choose K and so that

Ω Ω ¡ Ω εp εp χ φ ε # § ! T ! & 5 & !

K E and m K [ Ω wdm ] to obtain

E p

χ φ ε 5 & Z !

& p ] [ E L Ω wdm ].

Note that φ is inﬁnitely differentiable and with a compact support Ω, φ 1

on K, and 0 G φ 1. Here w is a nonnegative weight function, i.e., we refer to the

T © measure E E wdm. Proof: (We only sketch the proof; see [Adams, Section 2.17] for details on

molliﬁers.) ¡

¡ c

¥ Ω §g _C ¢ Ω ¥ §G

1 w ³ 1: Because r : d K 0, we can set K : x d x K

χ ∞ E

J H φ φ C φ H

r 2 , and then take : ´ for some large k, where converges U k K k c to the delta distribution δ0. The last claim follows from the regularity of m [Rud86, Theorem 2.20].

1 ¡

¢ l 6§ k l

2 The general case: Let w L Ω , where E Ω Ω and Ω is open.

p ¡

# § ! ε Ω

By [Rud86, Exercise 1.12], T Ω wdm , when m K is small enough.

S S

K Consequently,

χ p p

5 * G ! Q * φ ε E wdm wdm (B.9)

Ω Ω n

p ∞ p The space of simple L functions (as well as c ) is dense in L , even in

p1 p2 pn

µ';';'+ ! ∞

L L L (and even with different weight functions), when p :

∞ p n

! ∞ Theorem B.3.11 ( is dense in L ) Let Ω R be open and 1 G p .

p p ¡ ∞

(a1) Simple L functions are dense in L Q;B § , and countably-valued L

∞ ¡

functions are dense in L Q;B § .

∞ ε ¥;Q;Q;Q+¥ ¢_£ ¥ § g (a2) If we are given n ¢ N 1, exponents p1 pn 1 and 0,

n pk ¡ p

§ ¢ then, for any f ¢t L Q;B , there is a simple function s L s.t.

k 1

5 & Z ! ε & p

f s L k Q;B ] .

(a3) At least in (a1)–(b2) and (d), given a dense subspace B0 of B, we may

choose the (dense set of) functions so that they have their values in B0.

∞ ¡ p

§ Ω § Ω

(b1) Finite-dimensional c ;B functions are dense in L ;B .

¥ ¥RQ;Q;QzH ¥;QRQ;Q+¥ ¢£ ¥ § (b2) If we are given n ¢aC 1 2 , exponents p1 pn 1 ∞ , nonnegative

1 ¡

¢ § g ¢ ¥;Q;QRQR¥ Ω ε (weight) functions w1 wn Lloc ;B , and 0, then, for any f

n pk ¡ p

¥ § ¢ L Ω w dm;B , there are a simple function s L and a ﬁnite-

k 1 k

∞ ¡ Ω § φ

dimensional c ;B function s.t.

& 5 & ! ε & 5 φ & ! ε ¢¶C ¥;Q;QRQ+¥ HFQ Z

f s p Z and f p for all k 1 n ] ] L k Ω wk dm;B L k Ω wk dm;B

(B.10)

¡ ¡

¡ ∞ p

¢v· § ¢ §¨H § C φ φ

(b3) R;B c iR;B is dense in L R;B .

E¸

¡ ∞ § (c) The closure of c Q;B § in L is 0 Q;B when Q is metrizable. 922APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

∞ ¡

(d) The closure of simple measurable functions in L Q;B § equals

∞ ∞

¡ ¡ ¡

8C ¢ § §$¢ ¢ HFQ L Q;B § : f L Q;B there is a compact K B s.t. f q K for a.e. q Q

E (B.11)

¡ ¡

¡ ∞ ∞

§ §( We have Q;B § L Q;B when Q is metrizable. Moreover, L Ω;B

0 K K

¡ ¡

∞ ¡ Ω ∞ ∞ ∞ ∞

! §$ § ! L ;B § iff dimB , and LK Q;B L Q;B if dimB . (By using molliﬁers one could obtain further density results; cf. pp. 29–52 of [Adams].)

p ¡

Recall that simple means ﬁnite-valued, hence s is a simple L Q;B § function

n χ ∞ ¢ C H & & !

iff s ∑ x for some n N, x B, E , and s .

k 0 Ek k k k p

Proof: (a1) 1 Case p ∞: This follows from Lemma B.2.5(b2).

p ¡

! ∞ ¢ § ε g

2 Case p : Let f L Q;B and 0. By Theorem 3.13 of [Rud86],

&b& & 5 & ! ε J IC ¢

there is a simple function s : Q © R s.t. f B s p 3. Set K : q

: ¡

¡ χ p

H §"! ∞ & 5M¹ &i! ε J ¹ ¢ Q s q §©P 0 , so that m K and f f 3, where f : f K L . By

E p

χ «º

f n

² º

χ χ «º

∞ & 5 &! ε J as n ©» , hence g : f satisﬁes f g 4 for M big enough.

K f M

& G Note that & g ∞ M.

∞ χ By 1 , there is a (countably-valued measurable) function h ∑ b : k 1 k E

χ k ¢ K © B (b B, E K measurable and disjoint, its characteristic function k k Ek

¡ ∞ 5 & §"! for each k) s.t. & g h ∞ is arbitrarily small. Because m K , it follows that

¡ ε 5 & ! J ¥ §

we can take & g h p arbitrarily small, say min 4 1 (and, simultaneously

& G ! ∞ & h ∞ M 1 ).

p & By applying the (scalar) dominated convergence theorem to & h B (with

1 ¡

¢ § & 5 the constant function M 1 L K as the majorant), we see that K h

n χ p ¡ ε p n χ ε ε ! J § & 5 & ! J !

∑ b & dm 4 for some n. Thus, f ∑ b 3 4 . k 1 k E B k 1 k E p k χ k

(b1) By Lemma B.3.10, we may approximate each Ek above by some

∞ ¡

§ φ ¢ Ω k c to get n n

¡ χ χ

5 §,& G & & & 5 & ! J & φ φ ε

∑ bk Ek k p ∑ bk B Ek k p 4 (B.12) k 1 k 1

n φ ε ε

5 & ! J (by the Minkovski inequality), hence & f ∑ b 4 4 . k 1 k k p

(a2) Work as in (a1) for each pk. Let K be the union of “K’s” and let M be

5 & the maximum of “M’s”. Require & g h ∞ to be small enough for each pk. Let n be the maximum of “n’s”.

Ω 'R';'

(b2) Write K , where K K are compact sets (e.g., K : l l 1 2 l

¡ c

¢ * *,G ¼ ¥ §? J H ¢AC ¥ ¥;Q;Q;QzH

C x Ω x l d x Ω 1 l ), then, for some M 1 2 , we have E

pk pk ¡

χ χ Lº

& 5 ! ¥;Q;Q;Q+¥ §>¥ f f & w dm ε k 1 n (B.13) K f M k

Ω M B B ²

pk 1

& ¢ ¥;Q;QRQR¥

by the dominated convergence theorem, because & f wk L (k 1 n),

º χ χ «º by the assumption on f ; this gives the function g : f , where

K f B M ²

K : KM.

¡ pk

5 & ! ¥ § ! J & δ δ ε By taking g h ∞ min 1 , where K wk dm 4 for all k, we get a suitable countably-valued function h, and obtain then s as some partial sum of h, as in the proof of (a1). The rest goes approximately as in (b1) (when

B.3. LEBESGUE SPACES (LP Q µ;B ) 923

¥ ¥RQ;Q;Q¥ H applying Lemma B.3.10, use unions (over k ¢C 1 2 n ) of compact sets and intersections of open sets). (a3) This is obvious in the sense of simple (or countably-valued) functions. The procedure in (b1) obviously keeps the values in B0, so does that in (b2) too.

(b3) This is given in Lemma 2.3 of [Zimmermann] and its proof (which

¢ g shows that we may additionally require that 0 P suppφ if p 1). ¸ φ

(See Appendix D for · and the Fourier transform . Note that the claim ¸

follows easily from (b1) if p 2 and B is a Hilbert space.) (c) This follows from Lemma B.3.4. ∞ (d) (Note that LK equals the space L∞ used in the interpolation theory of

[BL].)

∞ ∞

1 Obviously, L , hence L when Q is metrizable, by (c). c K 0 K ∞ 2 SMF is a dense subset of LK: Let SMF denote the set of simple measurable functions (obviously, SMF L∞). K

ε ∞ ¢

Let g 0 and f LK be arbitrary. Choose K for f as in the deﬁnition of ¡

∞ n ε

¥;Q;Q;Q+¥ ¢ ¥ §

L . Choose n ¢ N, x x K s.t. K D x . 0 n

K k 0 k

1 ¡ k

£ ¥ §¦ # ¥RQ;Q;Q¥ 5 ε Set E : f % D x , E : E , E : E E (k 0 n 1),

k k 0 0 k 1 k 1 j 0 j :

n χ :

& 5 & ! & 5 & G s : ∑ x to obtain that s f ε a.e., hence f s ∞ ε. Since k 0 k E B

k ∞ ∞

¢ ε g 0 and f LK were arbitrary, we observe that SMF is dense in LK.

∞ ∞

¢ 3 LK is the closure of SMF: By 2 , we only need to show that f LK

∞

H & 5 & ! J ¢ assuming that C s SMF and s f ∞ 1 n for all n N 1 (so that L n n K is closed).

& 5 &½! J For each n 1, choose a null set Nn s.t. sn f 1 n on Nn. Set

c ε ε ε £ ¦ g g J & 5 &"!

N : ¾ N , A : f N B. Given 0, choose n 1 , so that s f n 1 n n

c n χ 0/ P on N . Write s as s ∑ x with E E for k j.

n n k 1 k Ek k j

¡ n

ε § V ¥ ε §RX & 5 &! ε & &! ε

Then A D 0 ¥ D x , because f x on E and f k 1 k k k

¡ c

# ε g on kEk § N. Because 0 was arbitrary, the set A is totally bounded (i.e., ε ε precompact), hence so is K A (use 2 in place of ), hence K is compact

(use, e.g., Theorem 9.4 of [Bredon] and the completeness of B). Moreover, ¡

¡ c ∞

§"¢ ¢ ¢

f q §$ limn sn q K for q N , hence a.e. Therefore, f LK. ¡

∞ ¡ ∞

§e § ! ! ∞ ∞ 4 LK Q;B L Q;B if dimB : If dimB , then we can take

¡ ∞ ∞

¥>& & § ¢ ¢

K : D 0 f ∞ for any f L to observe that f LK. ¡

∞ ¡ ∞

Ω Ω ∞ ∞

§ ! 5 LK ;B §q L ;B iff dimB : Assume that dimB . Let

Ω Ω n H

C E are disjoint sets of positive measure (in fact, we need not have R k as long as this property is satisﬁed).

The unit ball D1 of B is not compact, by Theorem 1.23 of [Rud73], hence H there is a sequence C x D without limit points (by Exercise 2.26 and

k 1 Theorem 2.37 of [Rud76], this is equivalent to noncompactness in any metric

χ ∞ ¡ c

¢ Ω § £ ¦¿ _C H space). Set f : ∑ x L ;B to obtain that f N x whenever n N k Ek k

c ¦ N is a null set; in particular, f £ N is not contained in any compact subset of B,

∞ ¢

hence f P LK. ¡

1 ∞ ¡ ∞

§mP § (N.B. the above example also shows that L L Ω;B L Ω;B ; e.g.,

K ¡

¡ 1

H §?! ¢ § n C ∞ Ω choose Ek s.t. ∑k m Ek to have f L ;B too.)

If µ is the completion of another measure µ , then the simple functions

constructed above can be chosen to be “µ -measurable”: 924APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

£75 ∞ ¥ ∞ ¦ ¢ ∑

Lemma B.3.12 Let X B or X . Let s k N xk Ek , where xk X and

¢ σ = Ek ¢¶ for all k N. If is the completion of another -algebra , then there

H =

C ∑

n U

are sets E s.t. x s a.e.

k k k N Ek (We omit the trivial proof.)

We generalize the Hölder inequality to the case r g 1: ¡

p ¡ q

& & G=& & & &

& & & G&G=& & & & & & & ¢ § ¢ § ¥ ¢

Lemma B.3.13 ( & f g r f p g q) Let f L Q;B , g L Q;B2 , p q

¦ & & GÀ& & & & ¢ ¥ ¢ & & GÀ& & & & ¥ ∞ 0 and bb2 B3 b B b2 B2 (b B b2 B2). Then f g r f p g q,

1 1 1

% % where r % : p q .

n ∞ ε

¥ ! g g Moreover, if Q R , µ m, p q and 0, then there is R 0 s.t.

τt ε n ∞

& ! ¢ * *g ¢ & f g r for all t R s.t. t R (if f 0, then we can allow p ; if

∞

g ¢ 0, then we can allow q ).

8 ¥ §

Thus, we may have, e.g., B K (B3 B2), B B2 (B3 K) or B B2 B3 .

C ¥ HJ G G C ¥ H Note that we may have r ! 1, but min p q 2 r min p q .

∞ ∞ & & Gc& & & & Proof: 1 Inequality f g r f p g q: If p or q , then this is

∞ C ¥ H ¥ ¢Á£ ¥ § O& &

obvious (and r min p q ), so we assume that p q 1 . Set F : f B,

¡ ¡

& §;J §RJ ¥ $ G : Â& g B2 , t : p q q, t : p q p, so that rt p rt q and 1

1 S

t % t 1. Then, by the Hölder inequality, we have

r r r r r r r r r

& & G& & & & G=& & & & G=& & & & ¥

f g F G dµ F G U F G F G (B.14) r t t rt t U r p q

& G=& & & &

hence & f g r f p g q.

& & G & & G 2 Finding R: Assume that f p 1 and g q 1. By Lemma B.3.7,

χ χ t n

& & ! ε J & & ! ε J ¢ * *,g there is R g 0 s.t. c f 2 and τ g 2 for t R s.t. t q. DR p DR q

&x! n Consequently, & f τ g ε for such t.

p p0 p1

¥ ¦

If p ¢££ p p , then L L L :

0 1 ¡

p0 ¡ p1

& & G Cw& & ¥|& & H

& & & G G Cw&Cw& & & ¥>&¥>& & & H H ¢ §b § G

Lemma B.3.14 ( & f p max f p0 f p1 ) Let f L Q;B L Q;B , 1

G G p0 G p p1 ∞. Then

1 θ θ

& G& & & & G Cw& & ¥|& & HF¥

& f p f p0 f p1 max f p0 f p1 (B.15)

p 1 p 1

% 0

θ where : .

p 1 p 1 1 % 0 Proof: The scalar case is Theorem 5.1.1 of [BL]; see also Theorem 4.1.2 and p. 27 of [BL]. (The deﬁnition of L∞ in [BL] coincides with the standard one in the scalar case, by Theorem 1.17 of [Rud86].) Apply the scalar case to

p0 ¡

& ¢ § n & f B L Q to obtain the general case.

Next we note that, roughly speaking, L is separable iff p ! ∞ and B is separable:

p ¡

! § Lemma B.3.15 Let B be separable and 1 G p ∞. Then Q;B is separable

n *

iff Q is at most countable. If Q R is measurable and dµ Ã* f dm for some

1 ¡ n p

§ § f ¢ Lloc R ,then L Q;B is separable.

B.3. LEBESGUE SPACES (LP Q µ;B ) 925 ¡

¡ ∞

DCP H ¥ §

If µ is as above, µ Q §iP 0 and B2 0 , then L Q µ;B2 is unseparable.

¡ ¡

¡ ∞ p ∞

§! § §

If 0 ! µ E for some measurable E Q, then L Q;B and L Q;B 2 2

are unseparable for unseparable B2. Proof: 1 Preparations: For each k ¢ N, we let Pk be the set of points in n k R whose coordinates are integral multiples of 2 % , and Vk the collection of

k k

';'R'

% % 2 u 2 boxes with corners at points of Pk. Set V : kVk, so that V is

countable and any open G Rn is the union of disjoint elements of V (see

[Rud86, 2.19] for details). Let S B be dense and countable.

χ p ¡ n

Ä _C E ¢ ¥ ¢ H ¥ § 2 The closed span of : x G x S G V is dense in L R µ;B : n E ε Let A R have a ﬁnite measure and g 0. By the Monotone Convergence

χ χ

E ε

¤C ¢ * *>! H & 5 & ! J Theorem, there is R g 0 s.t. AR : q A q R satisﬁes A AR 1 4.

n χ χ εE 5 & ! J Choose open G R s.t. & 4 (see (B.28) and use the fact that µ

AR G 1

*4! H

is absolutely continuous w.r.t. m on Cw* q R ).

¥ ¥;Q;Q;Qb¢ Write G as a disjoint union of elements G0 ¥ G1 G2 V, so that khiG χ ∑ , and use the Monotone Convergence Theorem to choose k ¢ N s.t. k N Gk

χ k χ χ k χ

5 & ! J & 5 & ! J

& ∑ ε 4. Then, ∑ 3ε 4. G j 0 G j 1 A j 0 G j 1

χ k χ ε & 5 & ! ¢ Given x ¢ B, we have x x ∑ for some x S. By A j 0 G j 1

linearity, it follows that the closed span of Ä contains that of simple measurable

p ¡ n functions, i.e., it is equal to L R ;B § .

χ χ p ¡

KC ¢ ¥ ¢ H ¥ § 3 The closed span of Ä : x G Q x S G V is dense in L Q µ;B : E n Apply 2 to the zero extensions of f and µ onto R .

p ¡

§ C ¢ ¥ ¢ H 4 © Q;B : The closed span of x x S q Q is obviously dense

q E

p ¡

¥ § CbC H ¢ H

in Q X (e.g., use the proof of 2 with V : q q Q ). E

∞ ¡

§ CP H ! J 5 ? Q;B2 is unseparable for uncountable Q and B2 0 : Let ε 1 2.

& &x Choose x ¢ B2 s.t. x 1. Then the -neighborhoods of no countable subset

∞ χ χ χ

¢ & 5 & P of can contain every x , q Q, because x x ∞ 1 for q q.

q q q U ¡

∞ ¡

¥ § ¥ §

6 L Q µ;B2 is unseparable: Let Q0 : Q, r0 : ∞. Given Qk rk ,

¡ ¡ ¡

¢ ¥ § ! §! § _C ¢ *)!

choose rk 1 0 rk s.t. 0 µ Qk 1 µ Qk , where Qk 1 : q Qk * q

: : :

H #

rk 1 , and set Qk : Qk Qk 1. Then Q k NQk, and the sets Qk are disjoint :

and: of positive (or inﬁnite) measure.

χ Å

& &

Choose x ¢ B s.t. x 1. For each E N, set f : x . Then

U 2 B E

2 n Æ EQn

5 & P ε & ∞

fE fE U 1 whenever E E , so that the -neighborhoods of no countable

∞ ¡

J ¥ §

set can contain every fE for ε ! 1 2. Thus, L Q µ;B is unseparable. ¡

p ¡

∞

! §! 7 L Q;B2 § is unseparable for unseparable B2 if 0 µ E for some

χ p

¢ H measurable E Q: The subspace C x x B of L is isometrically isomor-

E 2 E p phic to B2, hence unseparable. Therefore, so is L . n

Recall from Lemma A.3.1(a1)&(a2) that dimH means the cardinality of an

2 ¡ arbitrary orthonormal basis of H. We have dimL Q;H § dimH for inﬁnite-

dimensional H’s and most Q’s:

¡ ¡

2 ¡

¥ §(

¥ §$ §Ç

Lemma B.3.16 (dimL Q ¥ µ;H dimH) Assume that H is a Hilbert space.

¡ ¡

2 ¡ 2 2

§$ § § u Then dimL Q;H dimL Q u dimH; in particular, dim Q;H cardQ

P dimH ( dimH whenever Q 0, Q is at most countable and H is inﬁnite- dimensional). 926APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

n ¡

É* *

If Q R or Q ∂D is measurable, µ Q §ÈP 0, and dµ f dm for some

1 ¡ 2

¢ ¥ § ¥ § f Lloc Q m ,then dimL Q µ;H cardN u dimH ( dimH whenever H is

inﬁnite-dimensional).

C H

Proof: 1 Let xa a A be an orthonormal base of H (so that dimH A) Let ¡

2 ¡ 2

¥ § C H ¥ §

F be an orthonormal base of L Q µ . Then f x L Q µ;H is an

a f F a A

2 ¡ 2 §

orthonormal base of L Q ¥ µ;H (its closed span is L , by the density of simple ¡

2 2 ¡

§ §x u L functions) of cardinality of F u A. Thus, dimL Q;H card F A

2 ¡

¥ §

dimL Q µ u dimH. ¡

2 ¡

§ §

2 Since the set of simple Q functions is exactly cc Q , the set ¡

χ 2 ¡ 2

C H § §Ç

is an orthonormal base of Q . Consequently, dim Q cardQ, q q Q

2 ¡

§$

so that dim Q;H cardQ u dimH, by 1 .

2 ¡

¥ § 3 Let Q be as in the latter paragraph. By Lemma B.3.15, L Q µ is

2 ¡ § separable. It is obviously inﬁnite-dimensional, hence dimL Q ¥ µ cardN.

2 ¡

¥ §(

Consequently, dimL Q µ;H cardN u dimH, by 1 .

P G

4 By Lemma B.2.2, A u dimH dimH when A and A dimH

cardN. n

(See the notes on p. 947.) 1 B.4. THE BOCHNER INTEGRAL ( Q : L Q;B B) 927

1 - 3}Ë B.4 The Bochner integral ( Ê Q : L Q;B B)

Adde parvum parvo manus acervus erit. — Ovid (43 B.C. – 17 A.D.)

1 ¡ In this section we deﬁne the Bochner integral L Q;B §?© B and present the Bochner integral extensions of several more and less known Lebesgue integral results. A casual reader might wish just to have a look at Subsections B.4.1–B.4.3

and Theorem B.4.6 and then skip the rest of this section, just remembering that *R',* the Bochner integral is “the Lebesgue integral with ‘ &$',& ’ in place of ‘ ”’. By Theorem B.3.11(a1) and Lemma A.3.10, we may use the natural deﬁnition and density to deﬁne the Bochner integral:

Deﬁnition B.4.1 (Bochner integral) We recall that for simple functions s : n χ ∑ ¢

x (x B, E measurS able for all k) we have set k 0 k Ek k k

§?¢ Q sdµ : ∑ xkµ Ek B (B.16) Q k 0

1 ¡

T § The unique continuous extension of Q ' dµ onto L Q;B is called the Bochner integral.

1 T Let f ¢ L . Then f is called Bochner integrable and the integrand of Q f dµ,

which, in turn, is said to converge absolutely.

& GÌ& & Obviously, &¿T Q sdµ B s 1, hence the same holds for any integrable

1 ¡

¢ § function s : Q © B, by Lemma A.3.10. One easily veriﬁes that if f L Q;K

and f 0, then this coincides with the (Lebesgue) integral deﬁned in Section B.1.

¡ ¡

¡ 2

T T

T ∞

§ § 5 G G

Sometimes we write Q f t § dµ t : Q f dµ (e.g., Q t dµ t ). If a ¡

∞ ¡ §y { §d b G and µ m (or m E 0 µ E 0, i.e., µ is absolutely continuous w.r.t.

b ¡ b b a

T T T T T

§ ! 5

m), then we set f t dt : f dm : Z f dm. For b a we set : . ]

a a a b a b

`£ ¥ ¦ If B K (or “B 0 ∞ ”), then the Bochner integral is called the Lebesgue

integral, etc. By Theorem 11.33 of [Rud76], a function f : Q © K is Riemann integrable iff it is bounded and continuous a.e. It follows that Riemann integrable functions belong to L1. Moreover, the Riemann integral coincides with the Lebesgue integral. We shall not be using the Riemann integral. An equivalent way to define the integral is to define it for simple functions in the natural way and then use Lemma A.3.10 and Theorem B.3.11 to extend it to all L1 functions. This definition is used in [HP] and illustrated below:

¡ 1 1

¥ §

Lemma B.4.2 The Bochner integral is in L B , its norm is 1 (unless L

S S H

C 0 ); in particular,

&G & & & & ¥ & f dµ f B dµ : f 1 (B.17)

Q Q S

and the integral commutesS with bounded linear transformations:

¡ ¡

¢Í ¥ §;§¨Q T f dµ T f dµ T B B2 (B.18) Q Q

928APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

T T T Λ Λ Moreover, Q f dµ is the unique element of B satisfying Q f dµ Q f dµ for

Λ all ¢ B .

χ 1 ¢ ∑

Finally, if s k N xk Ek L , with the sets Ek being disjoint and measurable,

T § then sdµ ∑ x µ E . Q k N k k It follows that our deﬁnitions of integrable functions and the Bochner integral are equivalent to those of [HP], Section 3.7.

¡ 1 §

Proof: The BL L ¥ B claim and (B.17) follow from Deﬁnition B.4.1. If ¡

1 ¡

L P 0 , then there are x B 0 and E Q s.t. µ E 0 ∞ , and we

χ ¡ χ

&? r& & §$ r& & have &¿T Q x E dµ x Bµ E x E 1.

T T

We have T T for simple functions, hence for all L functions, by

Λ E Λ

¢ H ¢ continuity; (B.18). Elements C x B determines x B uniquely.

E n χ 1 © If s is as in the ﬁnal claim, then s : ∑ x s in L , by the Mono-

n k 0 k Ek

T T n tone Convergence Theorem, hence Q sn dµ © Q sdµ.

The standard results can be extended with ease:

Theorem B.4.3 (Lebesgue’s Dominated Convergence Theorem) Assume that

! © ¢

1 G p ∞, that the functions fn : Q B be measurable (n N), that the

¡ ¡

¡ p

§ ¢ £ ¥ ¦¨§ § ∞

limit f q : limn ∞ fn q exists a.e., and that there is g L Q; 0 s.t.

¡ ¡

§)& G § ¢ & fn q B g q a.e. for each n N.

p ¡ p

§ ©

Then f ¢ L Q;B and fn f in L . In particular, if p 1, then

T T

∞

limn Q fn dµ Q f dµ. n :

5 &

(This follows by applying the scalar LCD Theorem with Fn : Î& f fn B, ¡

p Ï p ∞ § F : 0, L G : 2g Fn. Obviously, this does not holds for p .) From the above and the Monotone Convergence Theorem applied to

N χ ¡ §

∑ f we obtain that if the sets E Q n ¢ N are measurable and disjoint,

n 1 Ek n

T T

nEn n N n & sides converges absolutely (i.e., with & f B in place of f ). Next we extend the standard deﬁnition of a product measure:

ν © £ ¥ ∞ ¦ ν © £ ¥ ∞ ¦ σ

u u Deﬁnition B.4.4 (µ u ) Assume that µ : 0 and : 0 are -

ﬁnite, positive measures on Q and R, respectively. By µ Ð we denote the smallest

¢Ñ¥ §¢Ñ= BH σ C

-algebra containing E u E E E . The product measure of µ and E

ν is given by S

¡ ¡ ¡ ¡ ¡

§ C ¥ §"¢ H§ ¢µ §>Q

ν § ν Ð µ u E : r q r E dµ E µ ν (B.19) Q E

ν ∞ ν ¡

u u By µ u : 0 we denote the completion of µ . By L Q R;B we µ Ð ν

Since in the deﬁnitions and results of this chapter we have assumed µ to be

ν ν

u u complete (as in [HP]), we shall use µ u (not µ ) on Q R. The basic properties

of this measure are listed below:

ν σ

(a) The measure µ u is -ﬁnite.

¡ ¡ ¡ ¡

§ C ¥ §"¢ H§ ¢µ ν § ν

(b) We have µ u E : µ q q r E d for all E ν.

R µ Ð E

(n ¥ k 1).

C ¢ ¥ §?¢ H ¢ (d1) If E ¢Ñ ν, then E : r R q r E is measurable for a.e. q Q,

µ Ð q E

r ¡

¢ ¥ §"¢ H ¢

and E : DC q Q q r E is measurable for a.e. r R.

E ¡

ν ¡ r ¢ §Ç ¢

(d2) If N is a null set, then Nq §d 0 for a.e. q Q and µ N 0 for a.e. r R.

¡ ¡

= § ¤

= 6§ Ò= 7§$ ¤ Ó

u u cl u ν

(e) (“cl Ð ν”) Let be the collection of ﬁnite unions of elements Ð

µ Ð

¢µ § ! g ν ∞ ε u of . Let E ν and µ u E . Then, for any 0, there is µ Ð

χ χ

& 5 &e! ¢ÍÓ ε

F s.t. E F . ¡

p ¡

§ ¢ ¥ § C H ¢ ∞ (f) If f L Q u R;B , p 0 , then simple measurable functions sn of

Nn χ

¢Í ¢ ¥ ¢

∑

u u

form s b and E F , b B (n k N)

n Ð

n k n k n k n k Ô k 0 En Ô k Fn k

∞

5 & © © s.t. & sn f p 0 as n .

NnU χ

¢ ∑

Note that each sn can be written as U s , where E and

k 0 E n k n k

n Ô k

(d2) follow from the Fubini Theorem (with f E).

¡ ¡

§>¥ ν §e! ∞ OC ¢ (e) 1 Case µ Q R : Set : E ν E the claim in (e)

0 µ Ð

¤ holds for E H , so that we only need to show that ν, i.e., that is a

0 µ Ð 0

Ó Q σ

-algebra (because, trivially, u ). This follows from 1 1 and

0 1 Q 2 .

χ χ χ χ χ

& 5 & Õ& 5 & & 5 1 Q 1 One easily veriﬁes that c c and E F 1 E F 1 E E U

χ χ χ χ χ

& 5 & G& 5 & Ò&

U U F F U 1 E F 1 E F 1, hence 0 is closed under complements

and ﬁnite unions.

¡ j

Q ¢Ñ ¢ § #

1 2 Let E j 0 j N and E j NE j. Set E : E j 1 E , so

j 1 k 0 k

: ¡

ν ∞ §"! u that E jE j and the sets E j are disjoint. Because µ u Q R , we have

χ χ χ χ

Å Å

& 5 &q! ε J ¢ & 5 & ! ε J U

U 2 for some N N. But 1 2 for some Ö

E j Ö N E j j N E j F ¢Ñ

F ¢}Ó , hence E 0.

2 General Case: Let Q Q , R R , where Q Q and

n N n n N n 0 1

g ¢~ §! ¢ ' ε ν ∞

R R . Given 0 and E s.t. µ u E , there is N N s.t.

Ð ν 0 1 µ χ χ χ χ

& 5 & 5 ! J ¢Ó & & ! Z

Z 2. By 1 , there is F s.t.

Ð ] × Ð ] E E × Qn Rn 1 E Qn Rn F 1 ε J 2, hence (e) holds.

(f) The ﬁrst claim follows from (e) and Theorem B.3.11. For the

C ¥RQ;Q;QR¥ H second claim, given n ¢ N, let E k 0 N consist of the sets

n k n

¡ ¡ ¡

C §¿# E ¥ ¥;Q;Q;Q¥ HbH ¥ § ¨¥ § C

E ¯ E s 0 1 N , and note that s q r s q r :

k s n k k s n k n n n

§ ¥ ¢ ¢

s r for all q q E (k N). n k n k ν

As in the scalar case, the norm of a µ u -measurable function may be integrated in any order, and for L1 functions the same applies to the function itself:

Theorem B.4.6 (Fubini) Assume that µ and ν are σ-ﬁnite, complete, positive

¢ § measures on Q and R, respectively. Let f L Q u R;B .

930APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

¡ ¡

T T

& ¢ £ ¥ ∞ ¦B§ & & ¢ £ ¥ ∞ ¦B§ & ν (a1) We have R f B d L Q; 0 and Q f B dµ L R; 0 .

1 ¡

T T T T

& ! & & ! ¢ § & ν ∞ ν ∞

(a2) If Q R f B d dµ or R Q f B dµd , then f L Q u R;B .

¡ ¡ ¡ ¡

1 ¡

T ν ¢ § § ¥ § § §

(b) If f L Q u R;B , then g q : R f q r d r and h r :

¡ ¡ ¡

¡ 1 1

S S S

§ § ¢ § ¢ §

Q f q ¥ r dµ q are deﬁned a.e. and satisfy g L Q;B , h L R;B and

Q ν ν

f dµ u gdµ hd (B.20)

Q Ð R Q R

Proof: (a1)&(a2) This follows from the classical Fubini Theorem (Theorem 8.8 of [Rud86]).

(b) This is Theorem 3.7.13 of [HP]. n

The rest of this section is rather technical. ν

In order to satisfy the µ u -measurability assumption of the Fubini Theorem

and its several important applications, we need to show that our function is

(product) measurable and study the relation between f : Q u R B and f : Q ¡

L R;B § . We start from basic facts:

Lemma B.4.7 Assume that µ : 0 ∞ and ν : 0 ∞ are σ-ﬁnite,

¥ ¦

complete, positive measures on Q and R, respectively. Let p ¢£ 1 ∞ .

¡ ¡ ¡

§ ¥R'§"¢ § ¢ ¢ ν

(a) If f L µ u ;B , then f q L R;B for a.e. q Q.

¡ ¡

¢ § ¢ § © ¢

(b) If f L Q;B , g L R;B2 , and B u B2 B3 is continuous, then f g

¡ §

L Q u R;B3 .

¡ ¡

¡ p

¥ ¢ §( §;§ u

¡ ¡

¡ p p

¦ `£ ¦@¢ §;§ £ ¦ `£ ¦ §

£ f g L Q;L R;B . Conversely, if f g as elements of L Q;L ,

¡ ¡

¥R'§$ ¥;'z§ ¢

i.e., f q g q a.e. on R for a.e. q Q, then f g a.e. on Q u R.

¡ ¡

¡ ∞ p p ¢ § ¢K£ ¥ § ¢ §;§ ¢

(d) Let f L Q u R;B and p 1 . Then f L Q;L R;B iff f ¡

p ¡ p

¢ § & & Z r& & Z Z

§ p p p

u u

] ]¥] L Q R;B . If f L Q R;B , then f L Q Ð R;B f L Q;L R;B .

c ©

Proof: (a) Let s f pointwise on N , where N Q u R is a null set and n

¢} ¢ P { P

∑

s b , where E , b B and k j E E for n n k n k Ð ν n k n k n j

k N En Ô k µ

¥ ¢

all n ¥ k j N.

¢ ¥R'§ ©

Let n ¢ N. For each q Q, the function sn q : R B is countably-valued.

§ ¢ #

But there is a null set N Q s.t. E is measurable for all q Q N and

n k q

§? ¢

all k ¢ N, by Lemma B.4.5(d1). Moreover, ν Nq 0 for a.e. q Q, say, for

¡ ¡

# k k k k k ¥;'z§© ¥;'§

q ¢ Q N , where N is a null set. Set N : N N . Then sn q f q

¢ # k k

and sn q ¥;'z§ is countably-valued and measurable for all q Q N . Thus,

¡ ¡

§ ¢ # k l

f q ¥;'z§"¢ L R;B for all q Q N , hence a.e. ©

(b) Let fn : Q © B and gn : R B2 be countably-valued and measurable ¡

¡ ν © # # § §

(n ¢ N), and let fn f on Q NQ, R NR, where µ NQ 0 NR . Then ¡

c c ν ¡R¡ c c © §# §;§$

u u u

fngn f g on NQ u NR, and µ Q R NQ NR 0.

¡ ¡ ¡

p ¡ p

¢ © ¥;'§R§x¢ §

L R;B § for a.e. q Q, and q f q L Q;L . The ﬁrst claim shows

¢ £ ¦@¢ §

that f L is independent of the representative of f L Q u R;B . However,

¡ ¡

¡ p p

¦¢ §R§ © § £ f L Q;L R;B may have a representative h : Q L R;B that is not 1

B.4. THE BOCHNER INTEGRAL ( Q : L Q;B B) 931

measurable Q u R B, see Example B.4.18; but if h is measurable, then h f

a.e. on Q u R, by the converse claim.)

¡ ¡ ¡

¡ ν

5 §y # §y 5 § §w u If f g 0 a.e., say on Q u R N, where µ N 0, then f g q 0

a ¡ p

¦1 `£ ¦§¢ § a.e. on R for a.e. q, by Lemma B.4.5(d2), hence £ f g L Q;L .

a ¡

£ ¦q £ ¦¢ ØC ¥ §¶¢

Conversely, assume that f g L and set N : q r R u

¡ ¡ ¡ ¡

§ÈP ¥ §¨H § § ¢ #

Q f q ¥ r g q r . Then f q g q a.e. on R for all q Q N , where

¡ ¡

§ ¢ #

µ N 7§ 0, i.e., ν Nq 0 for all q Q N . But ¡

ν ¡ ν § § ¥

µ u N : Nq dµ 0 (B.21)

i.e., f g a.e. on Q u R.

¡ § (d) By (b), we can consider f also as a function Q © L R;B (by redeﬁning

f on a null subset of Q; this affects f only on a null subset of Q u R).

p ¡

¢ § C H

1 Let f L Q u R;B . Choose simple measurable functions sn as in

¡ S

S p

§ ¢ Lemma B.4.5(f). It follows that sn : Q © L R;B is measurable (n N). But

p p p

sn sm p Z p sn sm d dµ sn sm p 0 (B.22)

Ð ] L Q;L ] B L Q R;B

Q R ¡

∞ p ¡ p p © § © §

as n ¥ m , hence sn converges in L Q;L to some g : Q L R;B .

¡ ¡

H §¤© § ¢ #

Replace C sn by a subsequence s.t. sn q g q for q Q N and

¡ ¡ ¡ ¡ ¡

¥ §i© ¥ § ¥ §m¢ # §© § u

sn q r f q r for q r Q u R N, where µ N 0 µ N . By

k ν

¢ #

Lemma B.4.5(d2), there is N Q s.t. N §© 0 for all q Q N . Set

k k k

N : N N .

¡ ¡ ¡ ¡

# k k ¥ §© ¥ § ¢ §© §

Let q ¢ Q N . Then sn q r f q r for a.e. r R and sn q g q

¡ ¡ ¡

p ¡ p

§i § §

in L R;B § , hence f q g q a.e., i.e., as elements of L R;B , Because

¡ ¡

¡ p

§;§ ¢

µ N l k 7§q 0, we have f g as elements of L Q;L R;B ; thus, f g ¡ p ¡ p

L Q;L R;B §;§ .

¡ ¡

S S

S p p

§;§ 2 Let f ¢ L Q;L R;B . Then, by the Fubini Theorem,

p p p p

& & & & & & & ¥ & ν

Z Z

f f d dµ f p Z dµ : f p p (B.23) ]¥] p B L R;B ] L Q;L R;B

Q R Q

∞ n

& & <& & ! Z hence then f f p Z p .

p L Q;L R;B ]¥] ¡

If f '¥;'z§ is continuous w.r.t. one argument and measurable w.r.t. the other, then

f is product measurable:

is a separable metric space and the open subsets of Q are measurable.

¡ ¡ ¡ ¡

If f : Q u R B is s.t. f q L R;B for a.e. q and f r Q;B for a.e.

¢ §

r, then f L Q u R;B .

¡ c

§$¢ § ¢ Proof: Let N R be a null set s.t. f '¥ r Q;B for r R : N . Let

1 1

¡ ¡ ¡

§$ ¥;'z§"¢ § ¢

Q Q be s.t. µ Q # Q 0 and f q L R;B for all q Q .

C H ¢

Let qk k N be dense in Q . For each k N, ﬁnd a sequence of countably- ¡

¡ c

∞

for some null set N R (use Lemma B.2.5(b2)). Set R : R N . We

k k 1 k % 932APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

require that

¡ ¡ ¡

¡ c

& ¥ §y5 §)& ! J § ¥ ¢ ¥ ¢ §

f qk r sk n r B 1 n 1 k n N r Nk (B.24)

C H ¢

(replace sk n n N by its subsequence, for each k N, if necessary).

ÎC ¢ ¥ §È! J §¨H #

Set Bk n : q Q d q qk 1 n , A0 n : B0 n, Ak 1 n : Bk 1 n

: :

¢ C H

j kB j n for all k N (the sets Ak n k N are measurable and disjoint, their

¥ §! J ¢

union is Q , and d q qk 1 n for all q Ak n).

Then the functions

¡ ¡

¡ χ

¥ § § §

sn q r : ∑ sk n r q (B.25) Ak Ô n

k N ν

form a sequence of countably-valued µ u -measurable functions that converge

∞ ε © ¥ §x¢ g u

to f on Q u R , as n . Indeed, given q r Q R and 0, choose

¡ ¡ ¡

ε ¡ ε J & ¥ §(5 ¨¥ §)& ! J ¥ 7§q! J § N g 2 s.t. f q r f q r B 2 for d q q 1 N 1 . Then, for

n g N, we have

¡ ¡ ¡ ¡ ¡ ¡

& ¥ §@5 ¥ §)& G=& ¥ §y5 ¥ §)& ¤& ¥ §@5 §,&

f q r sn q r B f q r f qk r B f qk r sk n r B (B.26)

ε ¡ ε

J J §!

! 2 1 n 1 (B.27)

¡ ¡

¢ ¥ §i! J G J §

(here k is chosen s.t. q Ak n, so that d q qk 1 n 1 N 1 and hence

¡ ¡ ¡ ¡

¥ §@5 ¥ §)& ! J § ¥ § & ε

f q r f qk r B 2; note that sk n r sn q r ).

¡R¡ ¡ ¡

¡ c

# § # § § §R§

u u

But Q R Q Q u R Q R R is a null set, hence f is

measurable. n

If Q is also a topological space, then µ is called regular if

¡ ¡ ¡

E E

C § Ù ¥ H© C § ¥ H

µ E § inf µ V V E V open sup µ K K E K compact (B.28)

E E (µ is outer regular if the former and inner regular if the latter condition is satisﬁed).

If Q Rn is open, then, by Theorem 2.18 of [Rud86], any locally ﬁnite Borel- measure on Q is regular; in particular, m is regular on Q. One often deﬁnes a measure from another one by using a weight function; the ν

“formula d f dµ” works also in the vector-valued case: S

ν ∞ © £ ¥ ¦

Lemma B.4.9 ( f dµ) Let f : Q 0 be measurable. Set ¡

ν ¡ § ¢µ<§>¥

¹ E : f dµ E (B.29)

E S

ν S ν ν ¹ Then ¹ is a positive measure on Q; let be the completion of . Then ν gd g f dµ (B.30) Q Q

∞ 1 ¡ ν £ ¥ ¦ ¢ ¥ § ¢ when g is measurable Q © 0 or g L Q ;B (equivalently, g f

1 ¡ § L Q ¥ µ;B ).

n ∞ ν T !

If Q R is open, µ m and f dm for compact K Q, then is K

regular.

©£ ¥ ¦ ∞ ν

Proof: 1 Case g : Q 0 : The claim on ¹ is Theorem 1.29 of [Rud86], ν

which also contains the claim on g provided that g is ¹ -measurable. By Lemma

£ ¥ ¦ B.3.12, general g : Q © 0 ∞ will do. 1

B.4. THE BOCHNER INTEGRAL ( Q : L Q;B B) 933 ¡

1 ¡ 1

2 Case g : Q B: By 1 , we have g L Q ;B g f L Q µ;B .

¡ ¡

§ § Λ 9 Apply 1 to Reg 9 and Img to cover the case B K. Replace g by g to

obtain the general case. ν ν

3 Regularity: By Theorem 2.18 of [Rud86], measures and ¹ are regular, n

Standard changes of variable can be applied without problems for measurable

functions that nonnegative or integrable:

1 ¡

§ Fg

Lemma B.4.10 (Change of variable) Assume that φ ¢ J1;R is s.t. φ 0 on

J1 (or that φ 1 0 and φ has at most a countable number of zeros), where J1 is an

φ ∞

£ ¦ `£ ¥ ¦ interval. Set J2 J1 . Let X B or X 0 .

p © Then a function f : J2 © X is measurable iff f : J1 X is measurable.

φ ∞ ∞ p φ p & r& p & G ¢££ ¥ ¦ ¢ p ¢ Moreover, & f ∞ f ∞ . Let p 1 . Then f Lloc f Lloc. If

ε φ ε 1 ε p φ p ! 1! g ¢ p ¢

% on J1 for some 0, then f L f L .

¡ S

1 S ∞ § © £ ¥ ¦

If f ¢ L J2;B or f is measurable J2 0 , then

¡ ¡ ¡ ¡

φ §;§ φ § f t § dt f s s ds (B.31)

φ 1 \ J2 [ J2 Here, as elsewhere, intervals are equipped with the Lebesgue measure. φ

By Lemma B.5.5, the functions f and f p have “same” Lebesgue points.

¥ ¦ φ g ¥ §

Proof: We shall replace J by £ a b J s.t. r 0 on a b (note

1 1

#iC ¢ φ § H

J1 : J1 t J1 t 0 is a countable union of such intervals, and it is

£ ¥ ¦F £¦£ ¥ ¦¦¦ α β φ

enough to treat J1). Let a b .

1 1 1

¢ £ ¥ ¦ £ ¥ ¦B§ J

φ α β φ % 1 Note that % ; a b and 1 r (recall that we use one-

¦ φ £ sided derivatives at endpoints), and % E is a Borel set for each Borel set

φ φ ε

¥ ¦ £ ¦ g

2 If N £ a b a null set, then so is N : Set R : max . Let 0 and

¡ ¡

§ §i!

assume w.l.o.g. that N a ¥ b . Now N V for some open V with m V ε

J R (see Theorem 2.20 of [Rud86]). Write V as the union of a countable

¡ ¡ ¡

φ ¥ § £ ¥ §¦B§

number of disjoint open intervals: V n N an bn . Then m an bn

¡R¡ ¡ ¡ ¡;¡ ¡ ¡

φ §;§;§G* 5 * ¥ §;§ φ £ ¦¨§G φ £ ¦¨§! ε

m φ an §>¥ bn bn an R m an bn R, so that m N m V .

φ £ ¦¨§$

Because ε g 0 was arbitrary, m N 0.

C H 3 Case f : J2 © X is measurable: Let sn be countably-valued and

measurable and s.t. sn © f a.e. Redeﬁne each sn on a null set so that they

become Borel-measurable; then still s © f outside some null set N J . n 2

Consequently, sn p φ f φ outside φ N , which is a null set, by 2 . But sn p φ is a countably-valued Borel-measurable function, by 1 , for each n, hence

f p φ is measurable.

1 φ φ

4 Measurability: the general case: Exchange the roles of and % to

φ φ φ

§ p obtain a converse for 3 . (Note also that f p is measurable iff f is measurable.)

φ ∞ & <& p & G 5 & f ∞ f ∞ : This follows from 2 .

∞ =£ ¥ ¦ 6 (B.31) for X 0 : This is contained in (15) on p. 156 of [Rud86].

1 1 1 φ & & £ ¦ 7 L and L claims: Apply 6 to f (for L we note that K J is loc B loc 2

φ £ ¦

compact for compact K J and % K J is compact for compact K J ;

1 1 2

1 φ φ moreover, , % are bounded on compact sets).

934APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

8 (B.31) for X B: If X K, then this follows from 6 (for Re f 9 ,

§ Λ Λ ¢ Im f 9 ); in the general case we replace f by f ( B ) and use Lemma B.4.2.

p p p

! & & ∞

9 L and Lloc claims: If p , this follows from 7 applied to f B. If

∞

p , then this follows 4 and 5 . ¡

φ E φ xPg ÀC ¢ §hg H 10 Case 0: Let f be measurable. Now G : t J1 t 0 consists of a countable number of disjoint open intervals, by LemmaE A.2.2. The

image of the null set J1 # G is a null set, so it follows from Lemma B.2.5(d1) φ

that f p is measurable. The proof of the converse implication is analogous (set ¡

E φ ∞ C ¢ §!8 H

G : t J1 % t etc.).

¥ ¦ Now for X É£ 0 ∞ , we can write the integral as a countable sum of

integrals, by the Monotone Convergence theorem; for X B the results follow

as in 6 –9 . n

2 ¡ 2 2 By Theorem B.3.2, L is a Hilbert space, hence L §R" L . The multidimen-

sional version of this goes as follows:

¥ ¥;QRQ;QH

Lemma B.4.11 Let H be a Hilbert space and n ¢tC 1 2 . Then, for each

¡ ¡ ¡ ¡

Λ 2 n S 2 n

§¨¥ § ¢ ¥ §;§

¢~ L Q;H K , there is a unique F L Q; H K s.t. ¡

Λ ¡ 2 ¢ §;§¨Q f F f dµ f L Q;H (B.32)

&«Ú G=& & G¤Û & &«Ú & Λ Λ Z

Moreover, Z 2 n F n 2 n .

] ]

L K 2 L K ¡

2 ¡ n

¥ §R§

Conversely, by the Hölder Inequality, each F ¢ L Q; H K deﬁnes ¡ ¡ 2 n

Λ Û §>¥ § & &

F ¢ L Q;H K , by (B.32). In Example B.4.13(c), we have F 2 n,

& &«Ú Û

F Z 2 2 1, hence the constant n is optimal. By Example B.4.13(d), we ] L K

& ∞ may have & F 2 if K is replaced by an inﬁnite-dimensional (Hilbert) space.

¡ n

¥ § © Proof: Let Pk ¢t K K be the projection Pk : x xk. By Theorem

2 ¡

¢ § = ¥ ¥;Q;Q;Q¥ ¥;Q;Q;Q+¥ Λ B.3.2, there are F1 Fn L Q;H s.t. Pk k Fk f L2 (k 1 n) and F1

Λ .

& r& & fÜ & Fk Pk . Set F : . to obtain (B.32).

Fn Ý

Obviously, F is linear and 0 F 0, hence F is unique. By the

Λ &xG_& & & &xG_& Λ & Hölder Inequality, we have & F . On the other hand, Fk , hence

2 2

& G & & n & F n Λ .

p The L norm of a measurable function f : Q © B is the supremum of

q ¡

§ * ¢ J J * φ φ φ Q f dµ , where L Q;B , 1 p 1 q 1. We can even take to be simple, and, with some extra assumptions, smooth:

Theorem B.4.12 Let B be a Banach space, let µ be a complete σ-ﬁnite positive

∞ 1 1

¢£ ¥ ¦

measure on Q, let p ¥ q 1 and p q 1, and let X be a norming subspace of ©

B for B. Let f : Q B be measurable. Then the following hold:

& & & G

(a) We have & f ∞ sup Λ f ∞ ∞ if X0 is a norming subset of B for

º º

Λ Λ X0 1 B. ² 1

B.4. THE BOCHNER INTEGRAL ( Q : L Q;B B) 935 S (b1) We have

φ & & *,G ∞ ¥ f p sup * f dµ (B.33)

φ ,Þ Q

q ¡

Ä< tÄ Ä ÒC φ ¢ § & φ & G φ ¢

where either Ä` 8Ä 1 or 2, and 1 : L Q;X q 1 & f E

1 n χ ¡

E ∞ Ä C ¢}Ä ¢ ¥ ¢ ¥ §"! H L H , : ∑ x n N x X m E for all k . 2 k 1 k Ek 1 k k

n E

T ∞ & !

(b2) If Q is an open subset of R , µ is regular, and K & f B dµ for compact

n 1 ¡ ∞

§ Är KÄ §1vÄ K R (i.e., f ¢ L Q;B ), then we can take : Q;X in loc 3 c 1

(b1). ¡

p ¡ p

§ Ä ÈÄ

(b3) If µ is the counting measure (i.e., L Q;B §w a Q;B ), then 2 cc 1

m φ ¢ (b4) In (b1)–(b3), we may require, in addition, that φ ∑ x , where m N, k 1 k k

φk is scalar and xk ¢ X, for each k.

£ ¥ ∞ ¦ `£ ¥ ∞ ¦

& G ¥

& g p sup φgdµ ∞ (B.34) ¾ φ ,Þ & φ 0 Q

1 φ ¢ where Ä is as in any of (b1)–(b3); we can even drop the requirement g L , 1 as well as the requirement g ¢ Lloc (in (b2)). ∞

(d) If p , then, in (b1)–(b3) above, we may let X be any normed space s.t.

º º

B2. ²

∞

For p ! , we must replace “ ” by “ ” in (B.33) for such a general X

(see Example B.4.13), but the supremum is still nonzero, i.e., f P 0

¢~Ä φ P φ Q f dµ 0 for some .

1 ¡

¢ § (e) We have f 0 a.e. if f L Q;B and f dµ 0 for all measurable E R, E

1 ¡ n

§ or f ¢ L R ;B and f dm 0 for all bounded measurable E R. loc E

q ∞ (The sets L and c are deﬁned for incomplete normed spaces exactly as for the complete ones.)

T φ ¢ φ The assumption f L is required to make the integral Q f dµ well deﬁned;

1 Ä KÄ

for f ¢ Lloc and 3 it is redundant. Clearly it is not needed for (B.34).

¡ ¡

¥ § D ¥ § In (d), we may have, e.g., X D B B2 , B X B2 or X K.

See Examples B.4.13 and B.4.14 for “counter-examples”.

& &ig Proof: (a) Clearly “ ” holds, so it is enough to assume that f M on a

Λ Λ Λ & &xG * *sg

set E Q of positive measure, and ﬁnd ¢ X s.t. 1 and f M on a 0

set of positive measure.

& WC ¢ §)&ßg ε H

Pick ε g 0 s.t. E : t Q f t M has a positive measure.

ε Λ ¢

Choose a ¢ E , Aε E for f and as in Lemma B.2.8(a). Choose X

* 0 0

E U

¡ ¡

Λ §)*,g_& §)&$5 ε J & Λ &xG

s.t. * f a0 f a0 2. Then (recall that 1)

¡ ¡

Λ ¡ Λ ε ε

§)*4g=* §)*R5 J g_& §)&$5 g

* f a f a0 2 f a0 M (B.35)

& & g for a ¢ Aε, hence Λ f ∞ M. 936APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

T φ φ ¢¶Ä & &?GD& & (b1) If f ¢ L and , then Q f dµ f p by the Hölder inequality;

φ p &"GM& & ¢ P & & ∞

trivially, we have &§T Q f dµ f p for f L (i.e., f p ). Thus, we only

!<& & G ¢Ä & &qg ! ∞ φ φ need to assume that 0 M f p and ﬁnd s.t. Q f dµ M.

We divide the proof of this fact into three parts.

ÄÎ _Ä yg §xg `C ¢

Case I — p ∞, 2: Take M M s.t. µ E 0, where E : t

¡ ¡

& §)&¤g BH ! §h!

Q E f t M . Because µ is σ-ﬁnite, there is A E s.t. 0 µ A ∞.

5 §;J

Choose a ¢ A and A : Aε A for : M M 2 as in Lemma B.2.8.

¡ ¡

ε ε

& &? §?g=& §)&$5 g ¢

Choose x ¢ X s.t. x 1 and x f a0 f a0 M . Then for t A ,

¡ ¡ ¡ ¡ ¡

¡ ε ε ε

§)& ¢ §¿5 ¥ §b § & §¿5 §)&?!

we have & f t B f a0 f a0 , and x f t x f a0 . Therefore,

¡ S

φ χ S J §¢ÍÄ : x µ A , and

A U 2 ¡

¡ 1

φ % ε &? § & &e& §)&(5 g Q & f dµ µ A x f dµ x f a0 M (B.36)

Q A U ¡

∞ ∞

Äà cÄ §q! & & Case II — p ! , 2: 1 Assume that µ Q , f p 1, and

k χ ¢ f ∑ b , where the sets Q , j N are disjoint. j 1 j Q j j

δ δ & &© gO& &5

For each j choose x j ¢ X s.t. x j 1 and x jb j b j , where : ¡

¡ k p 1

§;J & & % §! 5 1 5 M ∑ b µ Q 1 M, j 1 j j

φ k p 1 χ ¡ & & 5 § For : ∑ b % x , we have (because q p 1 p) that

j j Q S S j 1 j

¡ ¡

¡ q pχ p

φ §)& & & § r& §)& ¢ ¥ & t dµ ∑ b j Q j t dµ f t p 1 for all t Q (B.37) Q Q

j 1

∞ & φ & ∞

for q ! , and ∞ 1 as well for q . Moreover, S S

¡ ¡ ¡ p 1 ¡ p

φ % δ δ

&" & & § §? & & 5 § 5 §>¥ & f dµ ∑ b j x jb j µ Q j f p h 1 h (B.38) Q

j 1

¡ ¡ ¡

¡ k p 1 k p 1

δ % δ & & & &(5 § §"! & & §$ 5

where h § : ∑ b b x b µ Q ∑ b µ Q 1

j 1 j j j j j j 1 j j

T φ &eg M, hence & Q f dµ M.

¢ 2 By scaling and density (Theorem B.3.11), any f L will do in 1 , if

¡ ∞

µ Q §"! .

3 For the general case, let Q j NQ j, where the sets Q j, j N are

¡ ¡

DC ¢ ';';' & §)& ! H disjoint and µ Q j §?! ∞ for all j. Set E j : t Q1 Q j f t B j .

χ p E

& & g ¢ By the monotone convergence theorem, T Q E f B dµ M for some j N.

χ χ j

φ φ φ ¢Ä Use now 2 to ﬁnd E j for E j and E j f ; the same will clearly do

for Q and f . Ä KÄ Case III — Ä KÄ 1: This follows easily from case 2.

(b2) There is a nested sequence of open sets Ω Q with compact closure j

Ω

s.t. j N j Q. By the monotone convergence theorem, it follows that for

χ k χ

& Z g ¢Ä

& p ∑ ]

some j, we have f Ω M. Find g g Ω x s.t.

L j µ;B j i 1 i Ei 2

&xg & &xg M : I& g f dµ M. Set M : max x 0. f Ω j g i i

1 ¡

¥ §

Because f ¢ L Ω µ;B , we can ﬁnd, for each i, a compact K Ω and

j i j

& & ! 5 §;J

an open V Ω s.t. K E V and T f dµ δ : M M 2kM .

i j i i i B f g

Vi Ki

∞ ¡ χ χ

§ G G φ ¢ Ω φ φ By Lemma B.3.10, there is some i c j s.t. Ki i Vi . Set : 1 B.4. THE BOCHNER INTEGRAL ( Q : L Q;B B) 937

k φ

∑ x ¢}Ä It follows that

S S S S i 1 i i 2 k k

φ φ δ

&"Ò& &§5 & 5 & & & Ò& &§5 g Q & f dµ g f dµ ∑ g X f B dµ g f dµ ∑ 2Mg M Ω Ω Ω

j j Vi Ki j i 1 i 1

(B.39)

∞

!r& & & & ¢ (b3) 1 Case p : If M f ∞ sup f , then there is q Q s.t.

q Q

¡ ¡

§)& g & § &©g ¢ & &x & f q B M, so that f q x M for some x X with x 1. Thus, then

φ χ we can take : x.

¡ ¡

∞

! § C ¢ §½P H

2 Case 1 G p : Set A : supp f : q Q f q 0 . If A is

countable, then (b1) applies; obviously, Ä 2 cc 1 cc. Assume then

cC ¢ & §)&¤g J H

that A is uncountable. Choose n ¢ N 1 s.t. An : a A f a 1 n

C H

is uncountable (hence inﬁnite). Choose distinct elements a A . For

k k N n

¡ φ & &m § g J T each k, choose xk ¢ X s.t. xk 1 and f ak xk 1 n. Then Q m f dµ

m ¡ m

g J φ ¢ ¢

∑ f a § x m n, where : ∑ x c and m N is arbitrary,

k 1 k k m k 1 k ak c

∞ <& & hence then (B.33) f p.

(b4) The functions φ constructed in the proofs of (b1)–(b3) are of this form.

£ ¥ ¦ ¢~Ä (c) Finally, for g : Q © X, X 0 ∞ , we can choose φ to prove (B.34) 1 as above. By Hölder inequality, φg ¢ L may be dropped; by the trick used in

¡ 1

¢ §d! H ¢

“3 ” above (with E j : tC t Q f t j ) we may drop the assumption f Lloc. E

(d) Case I above applies here too (mutatis mutandis). For p ! ∞, we still

have the Hölder inequality (“ ”); naturally for B2 K this reduces to (b1)– (b3).

φ φ

¢ÁÄ ¢ Finally, if f P 0, then some ∑ b (here b B for all k)

k 1 k k k

T φ φ P

satisﬁes Q f dµ P 0, by (b4), hence then b : Q k f dµ 0 for some k.

φ φ n P Choose x ¢ X s.t. xb 0, and set kx.

We now show that part (d) does not hold for p ! ∞:

1 1

¥ ¦ G ! =£ ∞

Example B.4.13 Let Q 0 2 , µ m, 1 p , p q 1.

¡ ¡ ¡ 2 1 0 p 2

χ χ S

¥ § ¢ ¥ §;§R§

(a) B D K K , X K, f L Q; K K , Then

[ \§á [ \§á â 0 1 0 â 1 2 1

1 Y p φ & g & & Q

& f p 2 sup f dµ K2 (B.40)

º º

φ q Z φ Q ]ã

L Q;X q 1

& & g=& &«Ú Z

Thus, f f Z q 2 . ]

p L Q;K ]¨ K

Ú Ú

& & & & & &

Z Z Z

However, f f p Z 2 f q 2 , hence

]¥] ]ã ]

p L Q; K K L Q;K K

&«Ú & g=& &«Ú

Z Z Z

f q Z 2 f q 2 .

] ]ã ] L Q;K ]¨ K L Q;K K

χ 1 0

& & g & &

Ú Z

(b) Analogously, f f Z q 2 2 if we set f

[ \§á ] p ]¨

L Q;K K 0 1 0 0 â ¡

χ 0 0 p ¡ 2

§;§

¢ L Q; K .

[ \§á 1 2 1 0 â

n χ ¡ n

¢ ¥ §

[ \

k 1 n 1 n n n

α α

& & &¿T &qG & & G

by P e , we get f 2 Z n, f udm 1 when u 1,

\ ] n n L [ 0 n ;B 2

¡ n

¥ § where we can take B Ò X K for any nontrivial Banach space X (replace

Λ Λ w# C H f by f for some ¢ X 0 ).

∞ ¡

χ 2 ¡

§ ¢Ñ ¥ §

(d) Finally, if f ∑ P , B : N where P K B is deﬁned

[ \

k 1 n 1 n n 2 n 2

&¿T &GÀ& & & &«Ú α α

Z Z

by P e , then f udm u 2 Z , i.e., f 2 1,

] ]ã ] n n L R ä ;K L Q;K B2

938APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

Ú Ú

& & ∞ & b& å& &

Z Z Z

although f 2 Z . Note that f 2 f

]ã ]

L R ; K;B2 ]¥] L Q;B2 K 2

& ∞ æ f & 2 .

p 1 Y p

∞ & & & & Proof: (a) Let q ! . Obviously, f p 2, hence f p 2 . Let

q ¡ 1 0 0 0

§>¥ & & φ ¢ φ S

L Q;X S 1. Set P : , P : . Then á

q 1 á 2 â 0 0 â 0 1

& <& & G_& & ¥ & φ φ φ

q Z

\ ] P1 f dm X P1 dm X L [ 0 1 ;X : a (B.41)

Q 0

& GO& φ & Z

& φ q

\ ] by the Hölder inequality. Analogously, P2 Q f dm X L [ 1 2 ;X : b,

2 2 2 2 2 q q q

& G G G GD& & & φ φ hence Q f dm X a b . For q 2 we have a b a b q 1

φ 2 2 2 q q

& G g G

and hence &¿T Q f dm X 1. For q 2, the maximum of a b given a b

Y Y Y Y Y

2 Y q 2 q 1 p 1 q 1 p 1 p 2 2

% % ! ! § r& & % ∞ 1 is 2 2 2 2 2 f p. Finally, for q we

φ φ Û T & G ¥ &¿T & G ! r& & have & Pk Q f dm X 1 (k 1 2), hence Q f dm X 2 2 f p.

(b) This follows from (a).

(c)&(d) These can be proved as for (a) (for p 2 q this is obvious:

2 2 2

& G & & r& & & φ ∑ φ φ

f n 2 Z , as in (B.41)).

\ ]

L [ n 1 n ;X 2 %

1 ¡

¢ §

(e) 1 For f L Q;B this follows from Lemma B.2.8(b) (since

Λ Λ T T A f dµ A f dµ 0 for such A).

1 ¡ n

¢ § 2 If f L R ;B and f dm 0 for all bounded measurable E R, loc E

g n

then n f 0 a.e. for all R 0, by 1 , hence f 0 a.e.

[ R R %

Even if f is strongly measurable, “weak Lp” differs from Lp:

p p 2 ¡

{ P ¢

Λ ¢ Λ

¢ { { P P ¢ ¢ I § ¢

Example B.4.14 (Λ f ¢ L for all Λ f L ) Set H : N . Deﬁne f ¡

2 ¡

§ § ¢ &¨ ¥ ;& & ç¥& c& &

N;H by f k : ek. Then, for any x H, we have f x 2 x 2 x Hæ .

& É& & ¢ & & Thus, & Λ f 2 Λ for all Λ H , although f 2 ∞. (The same holds for

χ 2 ¡

¢ § ¹ ∑

f : e L R ;H .)

]

k N [ k k 1 k

: ¡

p Ï

§ © ¢ By The Hölder Inequality, the formula Q;B f ∑ F f K deter-

t Q t t

¡ ¡

q Ï p

§ © '¢è § mines a contractive map Q;B F ∑F Q;B . Usually, all elements

p ¡ §;

of Q;B are of this form: ¡

p ¡ q 1

§ Á § G ! ∞ %

Lemma B.4.15 ( Q;B Q;B ) Let Q be a set, 1 p and p

¡ ¡ ¡

1 p ¡ q 1

§; O «§ §; O § q % 1. Then Q;B Q;B and co Q;B Q;B (with equal

norms). ¡

p ¡ p

§ ¥ §

Recall that Q;B refers to L Q σ;B , where σ is the counting measure.

ç ¡

p ¡ q p p

§; K «§

Note that it follows that r S;B S;B for S Z, where r : r (see 1 Y r (13.2)).

Proof: 0 Remarks: We allow B to be an arbitrary Banach space; this

p ¡

¥ ¦ §

cannot be done in the case of, e.g., L £ 0 1 ;B ; cf. [DU]. For inﬁnite Q and ¡

¡ ∞

rC H § §

B P 0 , the closed subspace co Q;B of Q;B is not dense, hence there ¡

∞ ¡ 1

§; ¢ ¢P L§ is F ¢ Q;B s.t. F f 0 for all f co, in particular, F Q;B (a constructive example is given in Exercise 3.4 of [Rud73]). 1 B.4. THE BOCHNER INTEGRAL ( Q : L Q;B B) 939

q ¡

G G ∞ ¢Ñ «§ 1 Sufﬁciency: Let 1 p . Let F Q;B . By Lemma B.4.12(b3),

p ¡

§R & &;é Z É& &;é Z

¢A p q êã] F Q;B and F Q;B ]¥ê F Q;B (and the same holds with co in

place of if p ∞).

p ¡

! ∞ ¢ §;

2 Necessity, case p : For the converse, assume that F Q;B and ¡

¡ p

! ∞ π ¢Ñ ¥ë §;§ π

p . Since B Q;B (here x is x at t and zero elsewhere), we

t t

H ¢Í ¥ §$

have Gt : F t B K B .

¢ § © ∑

This function G : Q B satisﬁes F f t Q Gt ft for each f cc Q;B ,

*| r* *)G=& &b& & ¢ § by linearity. Since * ∑ G f F f F f for all f c Q;B , we have

t Q t t c

&;é Z G`& & ÄÃ Ä

& q ê G Q;B ] F , by Lemma B.4.12(b3) (with 2 cc). Consequently,

p ¡

G ¢ Q;B , by 1 . ¡

¡ p

§ § It follows that G F on the closure of cc Q;B , i.e., on Q;B . Thus, F

is of the required form. By density (see Theorem B.3.11(a)), G F.

¡ ¡

§ ¢ §;

3 Necessity for co Q;B : If F co Q;B and p ∞, then the above proof

¡ § applies, mutatis mutandis, and G F on co Q;B , by density (see Theorem

B.3.11(c)), hence F is again of the required form. n

The Minkovski Integral Inequality (not to be mixed to the Minkovski inequal-

ν ν & GI& & K& & &¿T & GMT & & ity & f g p f p g p) says that R f d p R f p d ; also this can be

extended to vector-valued functions

¡ ¡

§ ¥ § Theorem B.4.16 (Minkovski Integral Inequality) Let Q ¥ µ and R ν be com-

σ ν u plete, positive, -ﬁnite measure spaces. Equip Q u R with µ , the completion

ν ∞

G G of µ u . Let 1 p .

∞ © £ ¥ ¦

(a) If f : Q u R 0 is measurable, then

S S

ν ν ∞

& Z G & & Z G Q

& p p ] f d L Q ] f L Q d (B.42)

R R

(b) If B is a Banach space, f : Q u R B is measurable, and M :

¡ ¡

T ν ∞ ν

& Z ! § ¥;'z§ ¢ ¢ & p R f L Q ] d , then g q : R f q d B is deﬁned a.e., g

p ¡

& & G

L Q;B § and g p M, i.e., (B.42) holds.

& Z ν ! ∞ & p

If we did not make the assumption f L Q ] d in (b), we would have S S R to write

& Z G & & Z & ν

Z Z

ç 1 p p

] ]

f d f (B.43)

]ã ] f q L R X L Q X L Q X R R

or use some other trick to make the function in the left well deﬁned (here it is taken ¡

¡ 1

¢ C ¢ & ¥;'§,& ¥;'z§©P ν to be zero for those q, for which f q L ). The set q Q R f q d

χ E

∞ H

Z Z

is measurable by the Fubini Theorem, hence so is ç 1 , and thus

]ã ] f q L R X (B.43) can be proved in the same way as (b).

Proof: (a) By the Fubini Theorem, the inner integrals deﬁne measurable

¡ ¡

¥ §

functions; in particular, g q § : R f q r dr deﬁnes a measurable function

£ ¥ ¦ Q © 0 ∞ .

940APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

S S S

& & G J J b

If s 0 is a simple function and s p U 1, where 1 p 1 p 1, then

¡ ¡ ¡

¡ ν ¥ § § § §

S S gsdµ S f q r s q d r dµ q (B.44)

Q Q R

¡ ¡ ¡

¡ ν ν ¥ § § § §$G & & Z ¥ p f q r s q dµ q d r f L Q ] d (B.45)

R Q R

T T

& Ø& & G & & ¢

& ν

Z Z

p Z p p

] ]

hence R f d L Q ] g L Q R f L Q , by Theorem B.4.12 (p

£ ¦ 1 ¥ ∞ ).

¡ ∞

¢ 'R';'

(b) Find sets Q Q with µ Q §¤! ( j N) s.t. Q Q and

j j 1 2

S S S

T ν ∞

& & ! Q j NQ j. The assumption R f p d implies that

ν p ν p ∞ * & & * G & & ! ¥

Z X f d dµ V f p d (B.46) B L Q ]

Q R R

& ! & & & & G & & !

& ∞ ν ν ∞ T

by (a) (applied to f B) for p , and T R f B d ∞ R f ∞ d for

∞ T ν ∞

& & ! ¢ # § p . In particular, R f B d for a.e. q Q , where µ Q Q 0, and

g is deﬁned on Q , hence a.e. ¡

p ¡ 1

T ν ∞ § ¢ & & ! Because L Q ;B § L Q ;B for j N, we have f d dµ ,

j j Q j R B

T ν © hence g j : R f d : Q j B is deﬁned a.e. and measurable, by the Fubini

Theorem.

& & Z G p

Because g j g on Q j, also g is measurable. By (B.46), g L Q ]

T & Z ν ! ∞ n & p R f L Q ] d .

The following result allows one to take Laplace and Fourier transforms of

¡ p

R;L § functions componentwise (see Proposition D.1.13).

¡ ¡ ¡ ¡

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

T '§ § T § §

T T T

T ∞ '§ § § § § § § © £ ¥ ¦ Lemma B.4.17 ( Q f 'z§ r dµ Q f dµ r ) Assume that µ : 0 and

ν ∞ σ ¥ ¦

: = ,©ì£ 0 are -ﬁnite, complete, positive measures on Q and R, respectively.

¡ ¡ ¡ ¡ ¡

1 ¡ p

¢ §;§§ § ¢a£ ¥ ∞ ¦ § '§ § Let f L Q;L R;B L Q u R;B , p 1 . Then g r : Q f r dµ

p ¡

¢ §

exists a.e., and g KT Q f dµ L R;B .

¡ ¡ ¡

§ § ¢ Thus, then g r §m Q f dµ r for almost every r R. See also Exam- ple B.4.18.

Proof: (By using the Fubini Theorem, one easily veriﬁes that f 0 as

¡ ¡

1 ¡ p

§;§ §

an element of L Q;L R;B iff f 0 as an element of L Q u R;B , hence

¡ ¡

1 ¡ p

§R§b § ¢

L Q;L R;B L Q u R;B is well-deﬁned (we used the assumption that f

§ £ ¥ ¦ £ ¥ ¦ © u

L Q u R;B ; by Counter-example 8.9(c) of [Rud86], there is f : 0 1 0 1 ¡

1 ¡ p

¦@¢ £ ¥ ¦ £ ¥ ¦ §;§

R s.t. f is not measurable, but £ f L 0 1 ;L 0 1 ;B has a representative

¢£ ¥ ¦ £ ¦ =£ ¦

that is measurable, because f t §( 1 a.e. for every t 0 1 , i.e., f 1 .) ¡

¡ p

¢ ¢ § & & G By Theorem B.4.16, g r § exists for a.e. r R, g L R;B and g p

p ¡

& ¢ §

& f 1. Set F : Q f dµ L R;B .

¡ ¡ ¡

χ p ¡

¢ ¢½ § §y § § ¢ 1 Case f Eb, b L , E : Now F µ E b, g r µ E b r (r R),

hence F g. 2 Case f is simple: This follows easily from 1 , by linearity.

3 General case: Let fn f in L , as n ∞, and let fn be simple and

¡ ¡ ¡

T T

§ 'z§ §>¥ ¢

measurable for each n ¢ N. Set Fn : Q fn dµ, gn r : Q fn r dµ (r R), so that Fn gn, by 2 . 1 B.4. THE BOCHNER INTEGRAL ( Q : L Q;B B) 941

Because is continuous, we have Fn © F in L , as n . By Theorem S

B.4.16, S

¡ ¡ ¡

5 & <& 5 § 'z§ § & G & 5 & I& 5 & © ¥ & gn g p fn f r dµ p fn f dµ fn f 1 0 (B.47) Q Q

∞

C H

as n © . Therefore, some subsequence of gn converges to g and F a.e., n

hence g F a.e.

¢ §

The assumption that f L Q u R;B is necessary in Lemma B.4.17:

¡ ¡ ¡ ¡

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

T T

'§ § P § §

T T T T

'z§'z§ § § P P § § § § £ ¥ ¦ £ ¥ ¦ © Example B.4.18 [ Q f r dµ Q f dµ r ] Deﬁne f : 0 1 u 0 1 R as in

Counter-example 8.9(c) of [Rud86] (which uses the Continuum Hypothesis), and

£ ¥ ¦ £ ¥ ¦b© ³ ¥ ¢££ ¥ ∞ ¦

deﬁne h : 0 1 u 0 1 R by h 1. Let p a 1 .

£ ¥ ¦ £ ¥ ¦¿© ¢ I£ ¥ ¦

Then f is not measurable 0 1 u 0 1 R, but for each q Q : 0 1 , we

¡ ¡ ¡

¡ p

§ §?¢ £ ¥ ¦ § £ ¦1 `£ ¦

have f q §$ 1 h q a.e. on R, in particular, f q L 0 1 ;B . Thus, f h

¡ ¡ ¡

a ¡ p p

¥ ¦ £ ¥ ¦ §;§ £ ¥ ¦ £ ¥ ¦ §;§

as elements of L £ 0 1 ;L 0 1 ;B , even as elements of 0 1 ;L 0 1 ;B .

¥ ¦ ¥ § ¢

Even worse, for each r ¢8£ 0 1 , we have f q r 0 for a.e. q Q. Thus,

¡ ¡

§ ¥ § ¢£ ¥ ¦ ¢

T T g r : T Q f q r dq 0 for each r 0 1 , although Q f dm Q 1dm 1

p ¡

¥ ¦ § P

L £ 0 1 ;B , hence g Q f dm (cf. Lemma B.4.17).

¡ ¡ ¡

a ¡ p

¢ § ¢ § §y §

If f L Q;L , then there is h L Q u R s.t. h q f q a.e. on R (hence as p

elements of L ) for a.e. q ¢ Q (we omit the nontrivial proof). Thus, then Lemma

¡ ¡ ¡

§ §

B.4.17 can be applied to h, but not to f : the value of Q f q § r dµ q may differ

S S

everywhere from S

¡ ¡ ¡ ¡ ¡ ¡ ¡

§$ § §$ § § § f dµ § r hdµ r h q r dµ q (B.48) Q Q Q

(this equality holds for a.e. r ¢ R), as shown in the above example. If a continuous function has a L1 limit function on the left boundary, then this

function is L1 over the whole rectangle and also sideways:

¡ ¡ ¡

¡;¡ 1

¢ ¥ ¦ ¥ § § ¢ C H ¥ § §

u u

Lemma B.4.19 Let f r s a b ;B and f Z L r a b ;B ,

]

r sÐ a b

¥ ¥ ¢ ! !

a ¥ b r s R, r s, a b.

¡ ¡;¡ ¡;¡ ¡

¡ 1 1

If f t f r in L a b ;B , as t r , then f L r s u a b ;B and

¡;¡ ¡

¡ 1

§"¢ ¥ § § ¢ ¥ §

f '¥ c L r s ;B for a.e. c a b .

¡ ¡

¥ ¦ ¥ §

Proof: Being continuous, f is Borel-measurable on Q : r s u a b ,

hence Lebesgue-measurable on Q. Due to the continuity and convergence,

& ¥;'§)& ! ∞

1 Z¥Z

S S S S

M : sup S f t . By The Fubini Theorem,

\ ] ] t 1[ r s L a b ;B

s b b s

§ & & & & & & s 5 r M f B dm f B dm f B dm; (B.49)

Q r a a r ¡

s ¡ 1

T ∞

& ! ¢ ¥ § ¢ § in particular, r & f B dm for a.e. c a b and f L Q;B .

(See the notes on p. 947.) 942APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

d B.5 Differentiation of integrals (dt Ê ) “Chesire-Puss,” she began, “would you tell me, please, which way I ought to go from here?” “That depends a good deal on where you want to get to,” said the Cat. “I don’t care much where–” said Alice. “Then it doesn’t matter which way you go,” said the Cat. — Lewis Carroll (1832–98)

In this section we present Lebesgue points and a few results on differentiation of integrals. Also for vector-valued functions, almost all points are Lebesgue points:

1 ¡ n §

Theorem B.5.1 (Lebesgue points) Let f ¢ Lloc R ;B . Then, for almost all S n S

t ¢ R , we have

¡ ¡ ¡ ¡ ¡

¡ 1 1

% %

& §b5 §)& §@ § § Q

lim m Dr § f s f t B ds 0 and f t lim m Dr f s ds

Z Z

]

r 0 Dr t ] r 0 Dr t : : (B.50)

¡ n

Such t are called the Lebesgue points of f , and Leb f § R is the set of such t.

¡ n

§ KC b¢ L5 *|! §

(Here Dr t : t R * t t r, Dr : Dr 0 .) From (B.50) it follows that

¡ ¡

§)&G_& & G ∞ ¢ §

& f t f ∞ for all t Leb f .

¡ ¡

§ §(

Note that if f is continuous at t, then t ¢ Leb f . Obviously, Leb f ¡

¡ 1

§ ¥ ¢ ¥ ¢ Leb g § Leb α f βg for any f g L , α β K. loc

Proof: The ﬁrst claim follows from Theorem 3.8.5 of [HP] for t ¢ Dk, if we χ

replace f by Dk f (k ¢ N); the second claim follows from the ﬁrst for (at least) S

same t: S

¡ ¡ ¡ ¡ ¡ ¡

¡ 1 1

% %

& f t m Dr f s ds B m Dr f t f s ds B 0

Z Z ] Dr t ] Dr t (B.51) Let N D be the null set where the ﬁrst limit in (B.50) is nonzero or does

k k

¢ #

not converge. Then (B.50) holds for t R k NNk, hence a.e. (To be exact, [HP] gives the result for cubes, but it can be easily generalized to any nicely

shrinking sets, as in Section 7 of [Rud86].) n

From the above theorem we obtain

1 ¡ § Corollary B.5.2 Let J R be an interval and f ¢ L J;B . Then, for almost all

loc

S S t ¢ J,

t h t h :

1 : 1

¡ ¡ ¡ ¡

§ & §y5 §)& Q

lim f s § dt f t and lim f s f t ds 0 (B.52)

h 0 h t h 0 h t

¡ ¡

¡ t

§$ KT § ¢ §

In particular, if a ¢ J and F t a f s ds, then F J;B and F f a.e. n

a b 5~T (Of course, we have set T b : a .) d B.5. DIFFERENTIATION OF INTEGRALS ( dt ) 943

One easily verifies that a function F of the above form is locally absolutely continuous (see (scalar) Definition 7.17 of [Rud86]). However, unlike in the finite- dimensional (or Hilbert) case, there are absolutely continuous functions that are nowhere differentiable (however, this is not the case for reflexive spaces, a fortiori not in Hilbert spaces; see, e.g., [DU] or p. 125 of [KOS] for details).

1 ¡ n There is a method for uniquely choosing representatives for Lloc R ;B § “functions” so that these representative have all possible Lebesgue points:

1 ¡ n n

£ ¦w¢ § ¢ Lemma B.5.3 (Lebesgue representatiS ve) Let f Lloc R ;B . For each t R s.t.

¡ 1

§ 5 *

lim m Dr * f xt dm 0 (B.53)

r 0 Dr t ]

¡ ¡ ¡ ¡

§ § § §

for some xt ¢ B, we set L f t : xt; for other values of t, we set L f t : 0.

¡ ¡ ¡ ¡

§© § ¢ §

It follows that L f § t f t for all t Leb f , hence L f f a.e. and

¡ ¡

§ & &qGI& & £ ¦

Leb f § Leb L f . Moreover, L f f ∞ everywhere and L f depends on f

only.

¡ 1 ¡

Λ ψ

¦¥L£ ¦d¢ § ¢ ¢8í § ¢ Moreover, for all £ f g Lloc R;Y , Y , R , and every t R

we have

¡ ¡ ¡ ¡ ¡ ¡

§$ § §>¥ § §¨¥ Lψ f § t ψL f t Leb ψ f Leb f

(B.54)

¡ ¡ ¡ ¡ ¡ ¡

Λ §>¥ & Λ § §,&xG& Λ § §)&,¥

Leb f § Leb f L f t L f t

(B.55)

¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ α β α β α β §¿ §m ${ ¢ §>¥ §;§ §( § §>Q t ¢ Leb f Leb g t Leb f g L f g t L f t Lg t (B.56)

1 ¡ n n

§ ¢

¢ ¢ §LH

Bt : C x B t Leb LFx (B.57) E

is a subspace of B. n ¡

(We omit the simple proof.) Note that Leb L f § becomes the union of the

¦ © £ ¦©

Lebesgue sets of all representatives of £ f , and that f L f and f L f are not ¡

¡ n

linear, neither f L f t for any t R (but f L f and f L f are linear,

¦ `£ ¦ since £ L f f ).

The standard differentiation formula for integrals can easily be extended:

¡ ¡

1 Ï

¥ ¥ ¢ ¥ §Ç© §

Lemma B.5.4 Let J J R be intervals, a b J;J . Let f : J u J t s

b t ]

¡ ¡ ¡ ¡

¡ 1

¥ §"¢ ¥ ¢ § § ¥ § §

f t s B be s.t. f ft J u J ;B . Then F t : f t s ds is in J;B , Z

a t ]

and S Z

b t ]

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¥ § § ¥ §;§@5 § ¥ §R§ ¢ Q

F t § : ft t s ds b t f t b t a t f t a t for all t J (B.58) Z

a t ]

Recall that we allow J and J be non-open (the derivatives at the endpoints are,

of course, one-sided).

Z Z

] ]

b t b t ] a t

T T T

5 ¢

Proof: Because Z , where c J is arbitrary, we may

a t ] c c assume that a is a constant. Now

944APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

S Z

b t h ]

¡ ¡ ¡ ¡ ¡

§y5 §¿ ¥ §;§;§

F t h F t f t b t ds (B.59) Z

b t ] S

FromS the ﬁrst term we compute

Z Z ]

1 b t ] b t

¡ ¡ ¡ ¡

¥ §y5 ¥ §R§ 5 ¥ § ï

f t h s f t s ds ft t s ds

S S S S

h î

a a (B.60)

Z Z ]

b t ] t h t h b t :

1 : 1

¡ ¡ ¡

by the continuity of ft . (By the Fubini Theorem, we were allowed to interchange the order of integration.)

One easily veriﬁes that the second and third terms of (B.59) multiplied by

¡ ¡ ¡

§ ¥ §;§ 1 J h converge to 0 and b t f t b t , respectively. The continuity of F follows

analogously. n

A smooth change of variable preserves Lebesgue points:

∞ ∞ φ 1 ¡;¡ φ G ! GW ¢ ¥ § § eg ¢

Lemma B.5.5 Let 5 a b , a b ;R , 0 and f

¡ ¡ ¡ ¡

1 ¡;¡

α ¥ β § § α φ § β φ § §$ φ £ p φ §¦ Lloc ;B , where : a , : b . Then Leb f Leb f .

1 ¡ Here Leb refers to the zero extensions (or any other Lloc R;B § extensions) of φ

f and f p . ¡

¡ φ φ ¥ § p § g

Proof: Let T ¢ a b . Set g : f , t : T . Choose R 0 s.t. ¡

£ 5 ¥ ¦ α ¥ β § & φ & & φ &

\ [ \ t R t R . Set M : max [ , M : maxφ .

t R t R t R t R

% %

: : S

By LemmaS B.4.10, we have

φ 1 Z

t r t r ] :

1 : 1

¡ ¡ ¡ ¡ ¡

φ §@5 §)& & §y5 §)& §

& f s f t B ds g s g T B s ds (B.61)

φ 1 Z

2r t r 2r t r ]

% %

T M U r

MM :

¡ ¡

& §@5 §,& Q

g s g T B ds (B.62)

2rM T M U r

¡ ¡

§ ¢ §

But this converges to zero whenever T ¢ Leb g , hence then t Leb f .

¡ ¡ ¡

¥ § £ p §¦ §

Because T ¢ a b was arbitrary, we have φ Leb f φ Leb f . Exchange

1 1 ¡

§¦ p § φ φ φ £ φ %

the roles of f and g (and and % ) to obtain that Leb f Leb f ,

¡ ¡

£ p §¦ n

i.e., Leb f § φ Leb f φ .

The Mean Value Theorem is only true for R-valued functions (not even for C-

2 ¡ it

£ ¥ ¦ £ ¥ π ¦ valued or R -valued; e.g., set f t § : e , a b : 0 2 ); in the multidimensional

case it becomes a mere inequality:

¢ £ ¥ ¦ §

Lemma B.5.6 (Mean Value Inequality) Let a ! b, and let f a b ;B be ¡

¡ ξ § ¢ ¥ §

differentiable on a ¥ b . Then there is a b s.t.

¡ ¡ ¡ ¡ ¡ ¡

§y5 §)& G 5 §,& §)&G 5 § & §)&,Q

& f b f a B b a f ξ b a sup f x (B.63)

]

t a b

¡ ¡

§ ¢ ¥ §

If, in addition, f a exists, then there is ξ a b s.t.

¡ ¡ §

f b §@5 f a

¡ ¡ ¡

ξ 5 §,& G& §y5 §)& Q & f a B f f a B (B.64)

b 5 a d

B.5. DIFFERENTIATION OF INTEGRALS ( dt ) 945

¡ ¡ ¡ ¡ ¡

Proof: Let Λ ¢ X be s.t. 1 and f b f a f b f a B.

¡ Λ

Obviously, ReΛ f § b Re f , hence

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

Λ Λ Λ ξ ξ

§w5 §,& §§5 §d 5 § § §G 5 §)& §,& ¥ & f b f a B Re f b Re f a b a Re f b a f B (B.65)

by the classical Mean Value Theorem. The second inequality follows from the

¡ ¡ ¡ ¡ ¡ ¡ ¡

§Ç5 §Ç5 § §? §Ç5 §

ﬁrst applied to F t § : f t f a t f a (note that F t f t f a ). n

Lemma B.5.7 Let J R be an interval and n ¢ N. Then the following are

n 1 ¡

¢ §

(i) F : J;B ;

n ¡

¢ § (ii) F f for some f J;B .

Moreover, if (ii) holds, then F b f .

¡ ¡

¡ t

T T

§§ §) § ¢ By F f we mean that F t F a a f s ds for all t J and some (hence

all) a ¢ J. See Lemma B.7.6 for an analogous result for absolutely continuous functions.

Proof: One easily veriﬁes that (ii) implies (i). Given (i), ﬁx a ¢ J and set

¡ ¡ ¡

¡ t

Λ Λ

§ §§ 5 § 5 §

f : F , G t : F a a f . Then F G 0 on J, hence F G 0

¡ ¡

Λ Λ Λ Λ Λ

§ § ¢

on J and F 5 G a 0, hence F G on J; this holds for all B , n

hence F G. Therefore, (i) implies (ii).

¡ ¡

¡ 1

§ß¢ § ¢ ¥R'§

If f '¦¥ q Ω;B for ﬁxed q Q, and f z is measurable and has

1 ¡ 1

¢ § ¥R'§ Ω

a common L majorant for all z, then T Q f z dµ ;B with derivative

¡ ¡

T Ω

¢ §

Q fz z ¥;'z§ dµ ;B , by (c)&(d) below: G

Lemma B.5.8 Let Ω be a metric space, let Q be σ-ﬁnite, let 1 G p ∞, let

¡ ¡

§ ¥R'§ Ω ©

f : u Q B, and set F z : f z . Then we have the following:

¡ p

Ω §;§ ! ∞

(a) We have F ¢ ;L Q;B , if p and (1.)–(3.) hold, where

¡ Ω §"¢ § ¢

(1.) f '¥ q ;B for a.e. q Q;

¡ ¡

§ ¢

(2.) f z ¥;'§"¢ L Q;B for all z Ω; ¡

p ¡ ∞ Ω £ ¥ ¦¨§ & ¥;'§,& G ¢

(3.) there is g ¢ L Q; 0 s.t. f z B g a.e. for all z .

¡ ¡ ¡ ¡

¡ 1

Ω Ω T ¢ §;§ ¢ § § ¥R'§ ¹

(b) If F ;L Q;B , then F¹ ;B , where F z : Q f z dµ.

¡ Ω

¥;'§,& G ¢

and in (a) we also require that & fz z B g a.e. for all z .

¡ ¡

§$ ¥;'z§

In case (a) we then also have F z fz z , hence then (d) applies.

Ω ¡ 1 Ω §e¢ § ¢

(d) Assume that R is an interval, f '¥ q ;B for a.e. q Q, and

¡ ¡

¡ Ω p ¢ § §$ ¥;'z§

F ¥ G ;L , where G z fz z .

¡ ¡

1 ¡ p 1

§ h ¢ § § ¢ Ω Ω ¹

Then F ;L and F G (and F¹ ;B and F z

¥;'z§ ¢ T Ω Q fz z dµ if p 1) for all z . 946APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

(e) Claims (c) and (d) also hold with H in place of if Ω C is open.

¡ Ω

See Deﬁnition D.1.3 for H ;B § .

¡ ¡

§x© § ¢

¡ Ω p ;L § .

(b) This follows from (a) Lemma B.4.2.

¡ ¡

¡ z

§y5 ¥ §( ¥ § ¥ ¢ Ω (d) By Lemma B.5.7, we have f z ¥ q f a q a fz s q ds (z a ). Moreover, Ω is σ-ﬁnite, separable and metric, and open subsets of Ω are

m-measurable. Thus, fz is m u µ-measurable, by Lemma B.4.8.

p ¡

¢ &'& §

Fix z ¢ Ω. Given now h Ω, we have (here p refers to the L Q;B S norm and q ¢ Q is the corresponding dummy (i.e., dependent) variable)

z h

¡ ¡ ¡ ¡

1 ¡ 1

% %

V §y5 §RX5 §)& I& V ¥ §y5 ¥ §+X & & h F z h F z G z h f s q f z q ds (B.66) p S z z p z

z h

: ¡

1 ¡

¡ ¡ ¡ ¡ ¡

because & fz s q fz z q p G s G z p 0 as s z, and G s ¡

¡ ∞ 1 G ! ¢ £ ¥ ¦¨§

G z §,& p is bounded ( M ) near z, by continuity, and M L z z h .

p 1 ¡ p

¢ Ω §

Therefore, F b G exists in L . Consequently, F ;L .

¡ ¡

¡ p

¢ Ω § § ¥;'z§

¡ ¡ ¡

¡ 1

¥ §© J ¥ §5 ¥ §;§ ¢ ∞ % fz z q limn n f z 1 n q f z q for a.e. q Q, hence also fz satisﬁes condition (2.)). Therefore, we get the conclusions from (d).

(e) This is analogous to the 1 case (use a path integral and (b5) (and (b1) n for p 1) of Lemma D.1.2 instead of Lemma B.5.7).

p ¡ § By (B.50), The averages of any f ¢ L R;B converge to f pointwise a.e.; we also have the convergence in Lp:

p ¡

S ∞ ! ¢ § Lemma B.5.9 Let 1 G p and f L R;B . Then

1 r

¡ ¡

Proof: By the Minkovski Integral Inequality, we have (here the Lp norm S refersS to the variable t)

1 r 1 r

¡ ¡ ¡

& f t s f t ds p f t s f t p ds sup f f p 0

* *

r s 0 r s 0 \ s 1[ 0 r

We ﬁnish this section by a technical lemma: d B.5. DIFFERENTIATION OF INTEGRALS ( dt ) 947

1 ¡

¢ £ ¥ § § Lemma B.5.10 Let T g 0. If 0 is a Lebesgue point of f L 0 T ;B , then

T ¡

st ¡

T st

T ©

Proof: Because 0 se % dt 1, we have (B.70) for constant functions f .

¡ S Therefore, we may assume that f 0 § 0, i.e., that

¡ ¡

¡ ¡ ¡ ¡

¡ t 2 st

T T

§ ¢ £ ¥ ¦ § § & § & G ε §

Set F t : 0 f dm 0 T ;B . Then F 0 0, 0 s e % F t dt B g ¡

T 2 st ¡

facts and partial integration, one easily obtains (B.70). n

Notes for Sections B.1–B.5 As indicated in the proofs, many of the above results are known at least to some extent or in the scalar case. A further treatment on Bochner measurability, Bochner integral and vector-valued Lp spaces is given in, e.g., Sections 3.5–3.9 of [HP] and in [KOS], [DU] and [Yosida]; the monograph [Dinculeanu] treats same concepts from the Bourbaki point of view. The scalar case (the Lebesgue integral and measurability and Lp and spaces) is contained in most books on real analysis, such as [Rud86]. 948APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

- Ω 3 B.6 Vector-valued distributions ð ;B

No, my friend, the way to have good and safe government, is not to trust it all to one, but to divide it among the many, distributing to every one exactly the functions he is competent to. It is by dividing and subdividing these republics from the national one down through all its subordinations, until it ends in the administration of every man’s farm by himself; by placing under every one what his own eye may superintend, that all will be done for the best. — Thomas Jefferson (1743–1826), to Joseph Cabell, 1816

Here we brieﬂy present straightforward vector-valued generalizations of some basic scalar distribution results. We have written the other sections so that the reader may skip this section, but its contents give a deeper insight to some concepts needed in, e.g., Section B.7.

Throughout this section, B is a Banach space and Ω is an open subset of Rn.

¡ ∞

Ω § Ω § The space is not equal to c , although rather close. One tradition- ally uses a rather complicated topology; fortunately, for most applications one

does not need to know this topology, just some of its basic implications:

¡ §

Deﬁnition B.6.1 The test function space : Ω is the set of functions

∞ ¡ n

the standard (locally convex, complete, non E metrizable) test function topology

φ H φ ¢ [Rud73, 6.3–6.5], where a sequence C k converges to iff there is a compact

α α n ¢

k k

¡ ¡£¡

§ K § § The elements of : Ω;B : Ω ;B are called (B-)distributions.

As in the proof of [Rud73, Theorem 6.6], one can show that a linear mapping

¡ Ω φ φ φ

α α ∂α

We deﬁne the th weak derivative (or th distributional derivative) T ¢

of T ¢ by

α α α

¡ ¡ ¡ ¡ ¡

5 § § ¢ §;§>¥ ∂ φ § φ φ Ω °

T : 1 ° T D (B.72)

¡ ¡

I5 6§ in particular, ∂T φ § : T φ if n 1. Here we have used the standard multi- α

n n ¡ α 1 αn α

* * ¥RQ;Q;QR¥ § ¤';';'¨ index notation: α ¢ N , α : ∑ α , x x : x x , D : j 1 j 1 n 1 n α α

1 n d ∂

D ';';' D ; here D : and is the corresponding weak derivative.

1 n j dx j j ¡

1 ¡ Ω Ω 1

Ω p 1 ∞

¥ ¦

compact K ; note that L L for p ¢£ 1 ). is identiﬁed with the

loc

¡ ¡ ¡

φ φ Ω Ω Ω

is linear and injective. Similarly, a constant (function) b ¢ B is identiﬁed with the

A distribution with zero partial derivatives is a constant:

§ Lemma B.6.2 Let Ω be connected and T ¢ Ω;B . If δ jT 0 for all j, then

T ¢ B.

B.6. VECTOR-VALUED DISTRIBUTIONS Ω;B 949

Λ ¢ ∂ Λ Λ∂ Proof: If Λ ¢ B , then T and j T jT 0 for all j, hence

Λ φ α ¢ φ ¢ T T Λ K, by [Rauch, p. 256]. Choose 1 s.t. Ω 1 dm 1, and set

bT : Tφ1.

Then ΛbT ΛT φ1 αΛ Ω φ1 dm αΛ for all Λ B , hence ΛTφ

£¡

T ¢ § ¢ T 8T

αΛ T Ω φdm ΛbT Ω φdm for all φ Ω , Λ B , i.e., Tφ bT φ Ω bT φ

¢ n for all φ. Thus, T bT B.

By induction, we see that if the kth partial derivatives of T are zero, then T is

a polynomial of degree k: ¡

Ω Ω α ∂α § * *4 { Corollary B.6.3 Let be connected and T ¢ ;B . If k T 0,

β β ¢ ∑

then T β k 1 q bβ, where bβ B for all . n

° ° ²

∂

§ ¢

Corollary B.6.4 If J R is an open interval, T ¢ J;B , and T f

1 ¡ ©

Lloc J;B § , then there is a locally absolutely continuous function F : J B s.t. ¡

¡ t

§( §£T ¢

T F and F t F a a f dm when a J; in particular F f a.e. n ¡

¡ t

5 §$ § ∂

(This follows by deﬁning G t : a f dm and then noting that T G 0.)

¢ § Similarly, for any F ¥ f Ω;B s.t. ∂ jF f , the derivative D jF exists and equals f , as one can easily show by using mollifiers. Notes The contents of this section are well known, although it may be difficult to find any references; some other results on vector-valued distributions are given in [Treves]. See, e.g., [Rud73] or [Rauch] for the scalar case; most scalar results also hold in our setting with same proofs, mutatis mutandis (some existence results require the Radon–Nikodym property).

950APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES - k ¥ p

B.7 Sobolev spaces W Ω;B 3

The reason that every major university maintains a department of mathematics is that it’s cheaper than institutionalizing all those people.

In this section, we briefly generalize some facts about scalar Sobolev spaces to their vector-valued counterparts. Most of the time we follow the scalar representations in [Adams]. A casual reader might skip the definitions and other results and just read Lemma B.7.6 and Theorem B.7.4, since they suffice for most applications. Also other readers might wish to read first Lemma B.7.6 to get some

intuition to Wk p spaces.

G ∞ ¢ ¢ Throughout this section, B is a Banach space, 1 G p , k N, n N 1,

Ω Rn is open, and m is the Lebesgue measure on Rn.

α ¡

∂α 1 ¡ n 1 Ω § α ¢ ¢ Ω § Deﬁnition B.7.1 ( g) Let f ¢ L ;B and N . We call g L ;B the

loc α loc

S S

αth weak derivative of f (on Ω) and we write ∂ f g if

α α ∞

¡ ¡ ¡

5 § ¢ §;§>Q φ φ φ Ω ° g dm 1 ° f D dm c (B.73) Ω Ω

(By using linearity, projections, molliﬁers and a partition of unity, one could in

α ¡

∞ ¡

§ ¢ § φ ¢ Ω φ Ω ° fact show that (B.73) for c implies (B.73) for all c° ;X , where X is as in Theorem B.4.12(d). We omit the proof.)

α n α n α *

Here we have used the standard multi-index notation: N , * : ∑ , j 1 j α α

¡ α 1 αn α 1 αn d

§ Á';';' ';'R' x ¥RQ;Q;QR¥ x : x x , D : D D ; here D : is the (classical) 1 n 1 n 1 n j dx j jth partial derivative (use Deﬁnition B.3.3 with the other coordinates ﬁxed). If

1 n 1, we write ∂ : ∂ . (Outside this and previous section, we use the same

notation for weak and ordinary derivatives.) ¡

1 ¡ Ω Ω 1

§ © ¢ § Recall that f ¢ Lloc ;B means that f : B is s.t. f L K;B for each compact K Ω. The weak derivate is unique (as an element of L1 , that is, a.e.), loc

by Theorem B.4.12(d).

k ¡ α α

§ * *|G If f ¢ Ω;B and α k, then ∂ f D f on Ω, by partial integration. See also Theorem B.7.4 and Lemma B.7.6.

∂α Ω ∂απ π Ω

U It is obvious that if f g on , then Ω U f Ω g on for any open

Ω Ω.

Deﬁnition B.7.2 (Wk p) The Sobolev space Wk p is deﬁned by

α ¡

k p ¡ p p

C ¢ § ¢ * *,G H

W Ω;B § : f L Ω;B ∂ f L when α k (B.74)

¥ ∞ ¦ ¢ for p ¢££ 1 and k N, with norm

α 1 Y p α

p ¡

& & & G ! §¨¥ & & & &

& ∂ ∞ ∞ ∂ ∞

f : ∑ f 1 p f : max f (B.75) á

k p â k p α

α k k

° ° ²

¡ ¡

∞ ¡

Ω k p Ω k p Ω Ω 1 § § We denote closure of ;B § in W ;B by W ;B . When R ,

c 0

¡ ¡

k p k p

§ ¢ § we set f ¢ W Ω;B if f W J;B for each bounded open interval J Ω. loc

B.7. SOBOLEV SPACES Wk p Ω;B 951

0 p p k p

Ω Ω ¢ Ω §${

In particular, W L . Let be open. Obviously, f W ;B

k p k p k p

Ω § πΩ ¢ U f W ;B ; in particular W W .

loc

¡ α

Ω Ω k p Ω α ∂ π k p Ω k ¢ § * *sG ¢ k § U Let , f W ;B , k and f g. Then Ω U f W ;B

α ∞ ¡

π ¢ Ω §

∂ πΩ Ω

U U U

with the same norm, and U f g (for f c ;B the latter claim

k p ¡ Ω § follows by integration by parts; for general f ¢ W0 ;B by continuity; the

former claim follows from this).

¡ k p Ω k p

Theorem B.7.3 The spaces W ;B § and W0 are Banach spaces.

¡ ¡

∞ ¡

k p k p

∞ Ω §x Ω § Ω §

If p ! , then ;B W ;B is dense in W ;B , and

¡ ¡

k p n k p n §

W0 R ;B §$ W R ;B .

¡;¡ ¡;¡ ∞

1 1

§ § ¥ § § CP H However, 0 ¥ 1 ;B is not dense in W 0 1 ;B when B 0 (take an

absolutely continuous function whose derivative has a jump discontinuity).

k p p ∞ ∞ k p The space W is dense in L (p ! ), because W . However,

c ¡ 1 ∞ ∞

W R;B § -functions are continuous, by Lemma B.7.6, hence not dense in L .

k p

C H Proof: 1 The completeness of W : Let fn be a Cauchy sequence in α

k p p

*)G C H

W and let * α n. Then ∂ fn is a Cauchy sequence in L , hence there is

¡ ¡

p α p ∞ ¡ α ∞ q

n c c S the Hölder inequality impliesS that

α α α α α α ¡

° ° ° f fn : 1 ° D fn dm 1 D f0 dm : f (B.76) α

k p

* *,G Because α and φ were arbitrary, we have f0 ¢ W and ∂ f0 fα for α k.

k p

Clearly fn © f0 in W . k p k p 2 Being a closed subspace of W , also W0 is a Banach space. The last sentence of the theorem follows from the straightforward generalizations of [Adams, 3.15–3.19] (note that on p. 53 of [Adams] the condition ‘contained in

“Uk”’ should be ’contained in Uk but not in Uk 1’). n %

One can interpret Wk p as a closed subspace of ∏ Lp, hence it is

1 j Nn Ô k

² ²

separable if B is (cf. [Adams, 3.4]). Similarly, Wk 2 is a Hilbert space if B is.

¡ ¡

k k p

§ ¢ § & & G & & ψ ¢ Ω ψ ψ ψ

If , then W and Mk ª k (as a multiplication b b

operator on Wk p) and ψ f can be weakly differentiated by the Leibniz’ rule. For ¡

Ú /

¥ § & & à& & Λ ¢ Λ Λ Ω 0

Ú Z

Ô Ô

Z Z

B B we have Z k p k p (unless ), ]

Ω Ω

] ]¥] 2 W B ;W B2 B B2

∂αΛ Λ∂α k p ¢ moreover, f f for f W . An open Ω R has the cone property, if there is a ﬁnite cone C s.t. each point Ω Ω

x ¢ is the vertex of a ﬁnite cone C congruent to C. In particular, any ball x or cube or a product of such will do.

Theorem B.7.4 (Sobolev Imbedding Theorem) Let B be a Banach space, let

Ω n ∞ ! g

the open set R have the cone property, and let 1 G p , mp n, and

j ¢ N.

¡ ¡ ¡ ¡

j m p j j m p j

§ § § Ω § Ω Ω Ω

Then W : ;B ;B , and W ;B ;B , and these b 0 0 imbeddings are continuous.

952APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

¡ ¡

j m p n j n

§ § ¢ In particular, W : R ;B R ;B . Of course, f means that there

0 0

¢ À¹

is f¹ 0 s.t. f f a.e. See also Corollary B.7.7 (which allows p ∞).

α j j

& & & Proof: (We use the norm & f j : ∑ α sup D f in and .) By ª j b 0

° ° ² ¡ α

j m p

¢ § & & G Ω ∂

: ∞

[Adams, 5.4C], there is c c j m p n Ω s.t. for F W we have F

& * *,G & α

c F j m p when j.

¡ α

k j m p

¢ * *?G & § §)&Ñg α ∂ 1 Case f W : : Let j. If we had f t

α ¡

& ¢ & &iG §g & Λ Λ Λ∂

c f j m p : M, then there were X s.t. 1 and f t M. But

: ¡

α ¡ α

& G & & G & & ! § §

& ∂ Λ Λ ∂ Λ

f ∞ c f j m p c f j m p f t were a contradiction, hence

: : ¡

¡ α

§ §)& G & & & ∂

f t c f j m p.

j m p k

¢ C H

2 Case f W : : By Theorem B.7.3, some sequence f

j m p j

C H W : converges to f . By 1 , fn is a b Cauchy sequence, hence it j converges in b to a function g. Because a subsequence converges to f a.e. (generalize the corresponding scalar result, e.g., [Rud86, Theorem 3.12]), we

j have g f a.e. Moreover (because of the convergence in b ), we have

∂α α

& & & G & & & & Q

& ∞ ∞

f lim D fn lim c fn j m p c f j m p (B.77)

: :

n ∞ n ∞

j m p ∞ ¡ n

H § 3 W : Here we can take C f R ;X and obtain that g ( f )

0 n c

∞ j j

belongs to the closure of c in b , i.e., to 0 . n 4 The continuity of the imbeddings follows from the bound c.

If ﬁrst (resp. kth) weak partial derivatives of f are zero, then f is a constant

(resp. a polynomial of order ! k): ¡

1 p

Ω § δ Lemma B.7.5 Let Ω be connected and f ¢ W ;B . If j f 0 for all j, then

f ¢ B.

¡ α β

k p

Ω § * α * { ∂

Let, in addition, f ¢ W ;B . If k f 0, then f ∑ β k 1 x bβ,

° ° ² where bβ ¢ B for all β.

Λ Λ ∂ Λ Λ∂ ¢ Proof: If ¢ B , then f and j f j f 0 for all j, hence

Λ α ¢ f Λ K, by corresponding scalar result (see p. 256 of [Rauch]). Choose

φ φ φ T

1 ¢ s.t. Ω 1 dm 1, and set b f : f 1.

T φ Λ φ α α ¢ Λ φ

Then Λb f f 1 Λ Ω 1 dm Λ for all Λ B , hence f

Λ T ¥ Λ ¢ φ T φ T αΛ T Ω φdm b f Ω φdm for all φ B , i.e., f b f Ω b f φ for all

k p

¢ n φ. Thus, f b f B. The W claim follows by induction.

Being a W1 1 function on an interval is equivalent to absolute continuity:

1 p p p

T T

¢ §

Lemma B.7.6 (W L ) Let J R be an open interval and f L J;B .

Then the following are equivalent: ¡

1 p §

(i) f ¢ W J;B ;

¡ ¡

p ¡ t

§$ § ¢ § ¢

(ii) there is g ¢ L s.t. f t a gdm f a t J for some a J.

Assume (ii). Then g f a.e., ∂ f g, f is locally absolutely continuous, and

(ii) holds for any a ¢ J. If p 1, then f is absolutely continuous and has one-sided limits at the endpoints of J. B.7. SOBOLEV SPACES Wk p Ω;B 953

∂ ¢ ∂ Thus, if f ¥ f L , then the weak derivative f is also a pointwise derivative of f a.e. However, if f is the Cantor function of Theorem 7.16 of [Rud86], then

p ¡

¥ @¢ £ ¥ ¦¨§ ¢£ ¥ ¦ f exists a.e. and f f L 0 1 for any p 1 ∞ , but ∂ f does not exist (in

1 ¡ t

P §

Lloc), because f 0 and f f 0 0 0dm. Proof: 1 “(ii) { (i)”: Assume (ii). By using the Fubini theorem, one easily

∂ 1 p ¢

veriﬁes that f g, hence f W .

1 p p

{ ¢ ∂ ¢ ¢ § 2 “(i) (ii)”: Let f W and g : f L . Deﬁne F by F t :

t ¡

T § ¢ ∂

a gdt f a for some a J. By 1 , we have F g, hence F f b for

¡ ¡

§y5 §$

some b ¢ B, by Lemma B.7.5, and b F a f a 0. Thus, f F. Ç

3 Assume (ii). By Corollary B.5.2, f g a.e. and f is locally absolutely continuous. The rest follows from 1 –2 (recall that ∂ f is unique, by

Theorem B.4.12(d)), except the p 1 claims, which follow by choosing for

ε T ε ¢ G G<& & & &5 ! J g 0 a simple function s L s.t. 0 s g and J g sdm 2, and

δ ε δ ¡ δ J §q! setting ε : 2maxs (or ε 1 if s 0) — then m E ε implies that

ε ε & & ! g T E g dm ; because 0 was arbitrary, f is absolutely continuous (see,

e.g., Deﬁnition 7.17 of [Rud86]), hence it obviously is continuous on J¯. n

In case n 1, we can slightly improve Theorem B.7.4:

¡ ¡

¡ ¡ ¡ ¡

k 1 p k

§ §

§ § § § :

Corollary B.7.7 (W : J;B J;B ) Let J R be open. Then

¡ ¡ ¡ ¡

k 1 p k k 1 p k

§ § §m© § :

W : J;B J;B . The mapping W J;B K;B is continu-

ous for each compact interval K J.

k 1 p

¢ § ¢

Proof: Let f W : J;B . Let J J be an interval. Then f

¡ ¡

k 1 p k

§ ¢ §

W : J ;B , hence f J ;B , by Lemma B.7.6 and induction. Because J

k ¡ §

was arbitrary, we have f ¢ J;B .

¡ ¡

k 1 p k

§¶© §

By Lemma A.3.6, W : J;B K;B is continuous (because

k ¡ p

§ n

K;B § L K;B , continuously).

p For J R, the above weak derivatives are, in fact, L derivatives (and vice versa):

p ¡ §

Lemma B.7.8 Let f ¢ L R;B . Then the following are equivalent: ¡

1 p §

(i) f ¢ W R;B ;

¡ ¡

p ¡ t

§$ § ¢ § (ii) there is g ¢ L s.t. f t 0 gdm f 0 t R .

p 1 ¡ p

(iii) there is g L s.t. h % h I f g in L , as h 0;

¹ ¹

∂

If (i)–(iii) hold, then g g f a.e., f g, and f is locally absolutely ¹ continuous.

Proof: The equivalence (i) (ii) follows from Lemma B.7.6.

1 ¡ ∞

{ £ τ §Ç5 ¦ φ ¢ § 1 “(iii) (i)”: Set Dh : h % h I . Let c R be arbitrary. We

φ © φ © φ J J

S S S

1. Therefore,S by the Hölder inequality, we have the two convergences

¡ ¡

φ φ φ φ

§ 5 § © 5 ¥ g dm ñ D f dm f D dm f dm (B.78) ¹ h h R R R % R 954APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES

1 p

∂ ¢

¡ ¡ ¡

{ § £ §©5 §¦¨J

2 “(ii) (iii)”: Assume (ii). Set Fh x : f x h f x h

¡ ¡

1 h ¡

gral inequality (Theorem B.4.16), we get for ε g 0 that (here L norm is taken

w.r.t. x) S

¡ ¡ ¡

¡ 1

& §@5 §)& r& £ §@5 §¦ &

Fh x g x p : h S g x t g x dt p (B.79) 0

h ¡

1 ¡

& §@5 §)& ! ¥ G h g x t g x p dt ε (B.80) 0

*4! ©

when * h δε, by Lemma B.3.9. Therefore Fh g in L , as required.

3 Assume (ii). The identity g g f follows from 1 –3 (recall that ∂ f ¹

is unique, by Theorem B.4.12(d)), the rest from Lemma B.7.6. n 1 p 1 p The subset W0 of W refers to the elements that are “zero on the boundary”

in some sense:

1 p ∞ Lemma B.7.9 (W ) Let J R be an open interval and p ! . Then

¡ ¡ ¡ ¡ ¡

1 p 1 p 1 p

§ §Ñ C ¢ § §~ W J;B § J¯;B , and W J;B f W J;B f infJ 0

0 0

E ¡

f supJ §LH .

1 p 1 p ¢ Thus, a W J;B § function f has limits at endpoints of J; we have f W0 ∞ iff these limits are zero (they are necessarily zero at endpoints ò , if any).

1 1 ¡

¥ ¢Á£75 ∞ ¥ ∞ ¦ ¥ § Proof: 0 Let p q 1. Choose a b s.t. J a b . We shall

¡ ∞

! ¢ 5 ¥ § assume that a 0 b. By translation, this then extends to any a b .

¡ ∞

§ Using reﬂection, we can then cover intervals of form a ¥ . The case J R

follows from by Theorem B.7.4,

¡ ¡;¡

1 p p

§ ¢ ¢ ¥ § §

1 Let f ¢ W J;B . For n N 1, choose bn J s.t. L bn b ;B norm J of f and f is less than 1 n. By this, Lemma B.7.6 and the Hölder Inequality,

we have S

¡ ¡ ¡

¡ 1 1 q

§@5 §)&G & & G * 5 * ¥ ¢ ¥ §;§>Q & f s f t f B dm n t s s t bn b (B.81)

2 Assume that b ∞. By (B.81) and the Hölder Inequality we have

t 1 :

¡ 1

§)& 5 J G_& & G ' ¥ & f t B 1 n f B n 1 (B.82)

¡ ¡

∞ 1 ¡ ! ¥ §¢ § 3 Assume that b . By the Hölder Inequality, we have f f L J;B ,

in particular, f has a limit at b and f is absolutely continuous on J.

¡ ¡

1 p

4 By 2 and 3 , we have W J;B § J¯;B .

¡ ¡

5 Because f f b is continuous, by Theorem B.7.4, we have 0 f b

1 p ¡

∞ ∞

! §Ç

for f ¢ W0 for b ; for b this was shown in 2 . Analogously, f 0 0. ¡

1 p

6 Now only the converse for 5 remains: We assume that f ¢ W J;B

¡ 1 p

§ ¢

is s.t. f 0 § 0 f b , and prove that f W0 . By Theorem B.7.3, we may

¡ ¡

∞ 1 p

§¿ § assume that f ¢ Ω;B W J;B .

B.7. SOBOLEV SPACES Wk p Ω;B 955 ¡

1 p

¥ § ¢ Ω & 5

Let ε ¢ 0 1 . We shall construct g W with suppg and g

∞

& ! ε φ ¢ & φ 5 & ! ε Ô f W1 Ô p ; then one can ﬁnd c s.t. g W1 p by mollifying g (see [Adams]).

£ B¥ § x! ¢ δ

6 3 The case where f 0 on b b for some b b: Choose

¡ ¡ ¡

J ¥ J §;§ & & ¥|& & ¥>& & ! J ¥ §

0 ¥ min ε 5 b 2 s.t. f p f p f ∞ ε 5 over 0 δ . It follows that

¡ ¡ Y

1 Y q 1 q

§)&G J G J ! ¢ ¥ § ! & δ δ ε ε f 5 5 1 (cf. (B.81). Choose r 0 1 s.t. r % 2. Set

¡ δ ¢£ ¥ 5 § ¦

0 ¥ t 0 1 r ;

¡ ¡ ¡;¡

¡ 1 t

óô

§ δ δ δ

§ §>¥ ¢ 5 § ¥ §;§ g t : 1 5 r rδ f t 1 r ; (B.83)

¡ δ

¢£ ¥ §>Q

f t §¨¥ t b

ôö

¡ ¡;¡

1 p

δ δ δ δ

§;J 5 § ¥ §R§ It follows that g ¢ W and g f r on 1 r and g f on

¡ δ §

¥ b . Moreover, Y

p ¡ 1 q p 1 p p p

& ! J § J ¥ & δ δ ε δ ε

g p Z¥Z δ r 5r r 5 (B.84)

] ]

L 0 ;B

¡ ¡

& p

] ]

hence g L 0 δ ;B 2 5. Obviously, g f 5 on 0 , hence

& Z¥Z ! J & 5 & ! J J J J & ε ε ε ε ε ε

p 1 Ô p

] ] g L 0 δ ;B 5. It follows that f g W 5 5 2 5 5 ,

as required. ¡

5 ¥ § ! ∞ δ

6 3 The general case: If b , we can deﬁne g on b b in the same

5 & ! J & ε ∞ δ ε

way as above, so that f g W1 Ô p 2 . Let b . Then we can take : 6, ¡

1 Y q

$¢ ¥ § & & ¥|& & ¥|& & ! J

replace “b 5 δ” above by some b 1 ∞ s.t. f p f p f ∞ δ ε 5

∞ n § over b ¥ , and go on as above.

1 p p p

¢ ¢ H ¢

We set Wω : DC f Lω ∂ f Lω for ω R (the meaning of W0 is apparent

1 p ∞ 1 p from the context), and W ω denotes the closure of in Wω .

0 c

1 p α k p ©

Lemma B.7.10 (Wω ) The mapping Tα : f e f is a Banach isorphism of Wω

k p k p k p ∞ ∞ ∞

onto Wω α and of W onto W , and it is a bijection of onto and

0 ω 0 ω α c c

: onto ∞. :

Moreover, Theorem B.7.3, Corollary B.7.7 and Lemmas B.7.5, B.7.6, B.7.8

p p 1 p 1 p 1 p 1 p

0 0

1 1 ¡ § (except that if f ¢ Wω J;B , then T ω f is absolutely continuous, not necessarily

f ). %

j m p ω j

¢ ¢ Naturally, if f Wω , then e % f b as in Theorem B.7.4, etc. (but f

itself need not be bounded). ¢

¥ § ím X Y ; the above mapping does not map the derivatives of f to those of its

image Tα f .

ç ç ç

¡ ω ω ω

§ & &qG

Proof: 1 Bijections: Because e f ωe f e f , we have Tα f

¡ 1 1 p

*+ §)& & §L%

* α 1 f and Tα T α, hence the Wω claim holds. The claim on

∞ ∞ 1 p and is obvious, and the claim on W ω follows from these two. Use c 0

induction for general k.

2 Theorem B.7.3 and Lemma B.7.9: These follow directly from 1 .

¡ ¡

k p k p

¢ § ¢ §

3 Lemma B.7.5: Let f Wω Ω;B . Then f W Ω ;B for each open,

bounded Ω Ω, so the claim holds for k 1; use induction for general k.

956APPENDIX B. INTEGRATION AND DIFFERENTIATION IN BANACH SPACES ¡R¡

p ¡ p

¢ § ¢ 5 ¥ §Ç § 4 Lemma B.7.6: If f Lω J;B , then f L T T J;B for each

p ¢ T g 0, hence Lemma B.7.6 holds (we ﬁrst get g Lloc, but we must have

1 p p

¢ ¢ g ∂ f , hence f W iff g L ).

5 Lemma B.7.8 and Corollary B.7.7: Modify the original proof accord-

ingly. n

n p

The shift is a continuous operation on Wω :

n p ¡ n j j p

§ ¢ % § ¥ ¥;Q;QRQ+¥ ¢ τ

Lemma B.7.11 If f Wω R;B , then f R;Wω ( j 0 1 n). n

(This follows from Corollary B.3.8, Lemma B.7.8 and induction.) In some examples, we shall use following semigroups:

∞ ∞ ¡ ∞ 5 G ! ¥ § Proposition B.7.12 Let p ! and a b, and set J a b . If b

∞ π τπ p ¡ §

(resp. b ! ), then J J is a bounded C0-semigroup on Lω J;B , and its ∂ 1 p ¡

generator is the weak differentiation operator with domain Wω J;B § (resp.

¡ ¡

1 p ¡ 1

¢ § § H 5 § g C λ ∂ λ ω

with domain f Wω J;B f b 0 ) and its resolvent % (Re )

¡ ¡ ]

p Ï b λ t s

¢ § ©ØT §

maps f Lω J;B into the element J t t e % f s ds of this domain.

1 p ¡

In particular, for J R the domain is W ω R ;B . n

% % See, e.g., Examples 3.2.3 and 3.3.2 of [Sbook] for the proof (except for the last claim, which follows from Lemmas B.7.9 and B.7.10). By using R one obtains

the dual results for πJτ πJ. Notes The contents of this section are well known, although it may be difﬁcult to ﬁnd any references, particularly for the vector-valued case. Popular references for Sobolev spaces include [Adams] and [Ziemer] in the scalar case. Most of their results also hold in the vector-valued case with same proofs, mutatis mutandis.