Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

Norms, the Dual, Continuity Table of Contents Continuity The Evaluation Map The Evaluation Map Revisited Larry Susanka Convexity Schauder Bases in a Banach The Hahn-Banach Theorem Space Mathematics Program Norms The Banach Adjoint Bellevue College

June 13, 2013

Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

THE EVALUATION MAP THE EVALUATION MAP

E is linear, and also one-to-one: We suppose V is a and V∗ is its dual. that is, E(x) = E(y) exactly when x = y. V∗ is, itself, a vector space so it too has a dual, V∗∗ = (V∗)∗. There is an obvious collection of members of V∗∗, namely So if E is onto then it is invertible and an isomorphism. evaluation of a at members of V. In that case, (V∗)∗ can be identified with (i.e. it is) V. ∗ ∗ Define the evaluation map E: V → (V ) by If V is finite dimensional, V and V∗ have the same dimension. And it follows that (V∗)∗ has the same dimension as does V. So E(x)(f ) = f (x) for each x ∈ V and f ∈ V∗. E must be onto and therefore an isomorphism. The infinite dimensional case is much more delicate and we will consider the extent to which we can recover this important identification in some form later. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

CONVEXITY CONVEXITY

A nonempty subset S of a real vector space V is called convex if Geometrically, and in case V = R, this means that the graph of a convex always lies on or beneath the straight line tu + (1 − t)v ∈ S ∀t ∈ [0, 1] and u, v ∈ S. connecting any two points on the graph. For this reason convex functions are also called sublinear. In other words, all points on the line segment connecting u and v are in S whenever u and v are in S. So the region above the graph of such a function is a convex If V is any real vector space we say that a function P: X → is R subset of R2. convex provided

P(tu + (1 − t)v) ≤ tP(u) + (1 − t)P(v) ∀t ∈ [0, 1]. Any is convex: a seminorm is the most common source of convex functions.

Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

THE HAHN-BANACH THEOREM

Proof. Theorem If w ∈ X − Y and α, β are positive and u, v ∈ Y

The Hahn-Banach Theorem  β α  β Λu + α Λv = (α + β)Λ u + v α + β α + β Real Vector Real Vector Subspace Space  β α  If Y ⊂ X and P: X → R is convex ≤ (α + β) P (u − αw) + (v + βw) α + β α + β Λ ∈ ∗ Λ ≤ | ≤ β P(u − αw) + α P(v + βw). and YR satisfies P Y 1 1 So [Λu − P(u − α w)] ≤ [ P(v + β w) − Λv ]. ∃Ψ ∈ ∗ Λ = Ψ| Ψ ≤ then XR with Y and P. α β Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

Proof (Cont.) Proof (Cont.)

The left side does not depend on v or β, while the right is So Λ can be extended one dimension at a time while preserving independent of α and u. So there is a a with its relationship with P. 1 1 Let S be the set of all linear extensions of Λ to subspaces of X sup [Λu − P(u − α w)] ≤ a ≤ inf [ P(v + β w) − Λv ]. α β which are dominated by P on their domain. u∈Y v∈Y α>0 β>0 Partially order this set of extensions by Θ ≤ Ψ if Ψ is an L extension of Θ. Define Θ: Y Rw → R by Θ(v + rw) = Λv + ra for each r ∈ R and v ∈ Y. Chains in S have upper bounds in S and we invoke Zorn’s lemma and assert that there is a maximal member Ψ of S. Considering the cases of r positive, negative or zero separately, the definition of a yields The domain of Ψ is X, else it could be extended by one dimension, contradicting maximality. Θ(v + rw) = Λv + ra ≤ Λv + P(v + rw) − Λv = P(v + rw).

Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

THE HAHN-BANACH THEOREM:COMPLEX VERSION NORMS If you want to do calculus in your space you must have limits, Corollary and the easiest way to talk about limits in a vector space is through the explicit notion of distance provided through a The Hahn-Banach Theorem . If V is a vector space over or , a seminorm on V is a function Complex Vector Complex Vector R C If Y Subspace ⊂ X Space and P: X → satisfies R k · k: V → [0, ∞)

P(αv + βu) ≤ |α|P(v) + |β|P(u) if u, v ∈ X and |α| + |β| = 1 with the property that for any number k and vectors v and w

kkvk = |k| kvk and if Λ ∈ Y∗ satisfies |Λ| ≤ P| C Y and k v + w k ≤ kvk + kwk

∃Ψ ∈ ∗ Λ = Ψ| |Ψ| ≤ then XC with Y and P. The seminorm is called homogeneous by virtue of the first line. The second of these is called the triangle inequality. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

SEMINORMS NORMS

The triangle inequality can be tweaked slightly to produce a lower limit for the norm of a sum too. If you add the condition

| kvk − kwk | ≤ k v + w k ≤ kvk + kwk. kvk = 0 when and only when v = 0

A seminormed linear space, abbreviated SNLS, is a real or the seminorm is called a norm. complex vector space endowed with a seminorm. If G: V → F is any linear functional, the map |G|: V → [0, ∞) A seminorm satisfies given as |G|(v) = |G(v)| is a seminorm, and a common source of them too. This seminorm can never be a norm unless V has k αv + βu k ≤ |α| k v k + |β| k u k if u, v ∈ X and α, β ∈ F. dimension 1. normed linear space NLS So a seminorm is an example (the most important example) of a A , abbreviated , is a real or complex sublinear function as found in the statement of the vector space endowed with a norm. Hahn-Banach Theorem.

Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

METRICFROMA NORM CONVERGENCE The distance between vectors v and w in a SNLS V is defined by If a sequence v0, v1, v2,... converges in a SNLS using this d(x, y) = kv − wk. pseudometric we say that the sequence converges in seminorm (or norm). The distance notion is a pseudometric on V, and is a Sometimes this is also called strong convergence, particularly particularly nice one, having the properties when we have a norm from an inner product. d(x + z, y + z) = d(x, y) ∀x, y, z ∈ V (translation invariance) (There is a weaker concept of convergence which is also useful there.) d(a x, a y) = |a| d(x, y) ∀x, y ∈ V and a ∈ F (homogeneity) In case there might be confusion about the type of convergence not required of a general metric or pseudometric. involved, we might indicate intent by Whenever a notion of distance is used in an SNLS it is this vi −→ w. pseudometric which will be intended. strong This pseudometric is a metric exactly when the seminorm is a norm. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

NULL VECTORSFORA SEMINORM COMPLETENESS

If V is an SNLS the set N = { x ∈ V | kxk = 0 } is a vector subspace of V, sometimes called the set of null vectors for the Completeness is a very important property for us. Normed seminorm (not to be confused with vectors from the nullspace linear spaces which are complete are called Banach spaces. of a linear transformation.) R and C are themselves Banach spaces, a critical fact that is In an SNLS a sequence can converge to more than one point. used often and assumed without discussion in most first-year calculus classes. In fact, if vi → w then If S is a vector subspace of V then S is also a SNLS space, a vi → x exactly when x = w + y for some y ∈ N. subspace of V.

This implies that N is a closed set. If V is Banach, so is S. The seminorm is a norm exactly when { 0 } is closed.

Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

CONTINUITY CONTINUITY

For V and W both SNLSs, a function F: V → W is continuous at In fact if you can find δ1 > 0 for the single value ε = 1 and p = 0 p ∈ V exactly when then F is continuous. ∀ε > 0 ∃δ > 0 so that if k p − v k < δ then k F(p) − F(v) k < ε. That is because for any ε > 0 we have

kvk v 1 In general, δ will depend on both ε and p, but for linear kvk < δ ε ⇒ < δ ⇒ F < 1 ⇒ kF (v)k < 1 1 ε 1 ε ε functions it does not. Using linearity, one shows that continuity at any point, that is for just one v, implies continuity at every and so point, and the δ chosen for a particular ε does not depend on kF (v)k < ε. which point. Linear functions between two SNLSs Linearity then allows us to translate the argument away from which are continuous at a point are uniformly continuous. the origin. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

CONTINUITY BOUNDEDNESS

For FLinear : VSNLS → WSNLS we define kFk by A useful equivalent condition, again using linearity of F, is that k k = { k ( )k | ∈ ( ) }. if F−1(B) is open for even one open ball B then F−1(B) will be F sup F x x S1 0 open for every open subset B contained in W and therefore F If kFk is finite we say F itself is bounded, and it turns out that will be continuous. kFk is bounded exactly when F is continuous. This is important One implication: for continuous linear F, the set Ker(F) is a enough that we enshrine the result in: closed subspace of V. A linear functions between two SNLSs is continuous exactly when it is bounded.

Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

OPERATOR NORM THE CONTINUOUS DUAL

The collection of bounded linear functionals on an NLS V is a B(V, W) is defined to be the set of bounded linear functions particularly important (with operator norm.) from V to W. It is called the continuous dual of V, denoted The number kFk given for each F ∈ B(V, W) defines a norm on ( , ) 0 B V W , generally called the operator norm. V = B(V, F). If the range space W is a Banach space, so is B(V, W). The continuous dual is a subspace of the algebraic dual, If there are multiple norms floating around, we might k · k . ∗ distinguish this one by the notation op V = HomF(V, F). Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

00 EVALUATION IS INTO V HAHN-BANACHANDTHE CONTINUOUS DUAL If V is an NLS, then V0 is Banach. It too has a an algebraic dual Suppose x is nonzero in NLS V. The linear transformation Λ 0 ∗ 00 0 ∗ (V ) and continuous dual V ⊂ (V ) . defined on Fx by Λ(a x) = a kxk satisfies the condition of the Recall the evaluation map Hahn-Banach Theorem (or its Corollary) where P is the norm on V. In this one dimensional case we have equality: E: V → (V∗)∗ given by E(x)(f ) = f (x). | Λ(a x) | = |a| kxk = ka xk. Every member of (V∗)∗ produces a member of (V0)∗ by restriction and Ee(x) defined to be E(x)|V0 is such a member. So Λ can be extended to linear Ψ defined on all of V and for which And if f ∈ V0 and kf k = 1 then |Ψ(v)| ≤ kvk for all v ∈ V.

Ee(x)(f ) = |f (x)| ≤ kf k kxk = kxk. This means that the operator norm kΨk cannot exceed 1. But Ψ(x/kxk) = Λ(x/kxk) = 1, So kΨk = 1. So each Ee(x) is bounded as a function from Banach space V0 To recap, for each x ∈ V there is a functional Ψ in V0 with with operator norm to F. In other words, 00 kΨk = 1 and Ψx = kxk. Ee : V → V .

Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

REFLEXIVITY OVERVIEW

But by the result of the previous slide there is a functional Ψ ∈ V0 with operator norm 1 for which |Ψ(x)| = kxk. This

( ) k k means that Ee x actually attains its maximum value, x , on The Baire category theorem implies that no complete infinite 0 the members of V with operator norm 1. dimensional space can have a countable basis. However, if we Even more, this means that can’t have a countable basis, we can do almost as well with a Schauder basis, defined below, which uses concepts of limit 00 Ee : V → V is an isometry. and continuity provided by a norm to get most of what a true basis provides in the finite dimensional setting. So the image of Ee with operator norm and V itself with its norm are not only isomorphic as vector spaces but are The ideas to follow make sense in more general settings but we interchangeable in any calculation involving norms as well. will confine consideration here to Banach spaces. Spaces for which Ee is onto V00 are very important, and are called reflexive. We will have occasion to refer to this property later. Hilbert spaces are reflexive. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

SCHAUDER BASIS THE COORDINATE FUNCTIONALS X A Schauder basis for Banach X is a countable ordered set of When referring to a sequence of vectors in vector space the notation v = (vn) ⊂ X will be used. Thus, for Schauder basis as vectors v0, v1,... for which every member x of X can be written in a unique way as above we have the paired sequences ∗ ∞ v ⊂ X with unique associated coordinate functionals a ⊂ X . X n x = a (x) vn n=0 Any Banach space with a Schauder basis is separable, so there n are Banach spaces without Schauder bases. In fact, there are for a (x) ∈ F. The uniqueness refers to the values of the coordinate functionals an, which are therefore linear, and the Banach spaces for which no infinite dimensional subspace has a convergence of the sequence of partial sums is in norm: Schauder basis. Still, Banach spaces that have these bases are common in practice. k X n Uniqueness of coefficients implies that 0 is not among the a (x) vn −→ x ∀x ∈ X. strong vectors in a Schauder basis, and in fact the vectors in a n=0 Schauder basis must constitute a linearly independent list of vectors.

Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

CONTINUITYOFTHE COORDINATE FUNCTIONALS UNCONDITIONALAND BOUNDED BASES

If (vn) is a Schauder basis, so is (vn/kvnk) and the latter Schauder basis is called normalized. A Schauder basis is called unconditional if, for any x ∈ X the Normalized or not, the members of the sequence of linear series obtained by any permutation of the terms in the series functionals (an) are all continuous. In fact, (though it takes a bit representation for x in this Schauder basis also converges to x. of work to prove) their norms satisfy A Schauder basis is called bounded if there are positive n constants A and B for which 1 ≤ ka k kvnk ≤ K ∀n ∈ N A ≤ kv k ≤ B ∀n ∈ . for a positive constant K that will vary with the basis. n N So for Schauder basis v ⊂ X we actually have a ⊂ X0, not just a ⊂ X∗. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

DUAL SCHAUDER BASES THE BANACH ADJOINT Since each vector in a Schauder basis v can be conceived of as Suppose F ∈ B VBanach, WBanach. (that is, it is) a member of X00 the possibility arises that a could w0 ∈ W0 F∗(w0) ∈ V∗ be a Schauder basis for the Banach space Span(a) with operator Define for each the member given by v ∗ 0 0 norm, with coordinate functionals . F (w )(v) = w (F(v)). This is, in fact, the case. Even more, we have the following. Note that kF∗(w0)(v)k ≤ kw0k kFk kvk. The Dual Basis Theorem So, in fact Schauder k ∗( 0)k ≤ k 0k k k Suppose v Basis ⊂ XBanach with coordinate functionals a ⊂ X0. F w w F Schauder Then a Basis ⊂ Span(a) with coordinate functionals v. and this means F∗(w0) is actually in V0, not just V∗. Moreover, if v is unconditional or bounded, so is a. Looking again we see that F∗ ∈ B (W0, V0) and kF∗k ≤ kFk. The most interesting case, of course, is when X is reflexive. Application of the Hahn-Banach theorem proves more: that 0 Then X = Span(a) and so we have a correspondence between kF∗k = kFk. Schauder bases with their coefficient sequences for X and those It is also a fact that if F is an isometry, so is F∗. for X0.

Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint

THE BANACH ADJOINT

The map F∗ : W0 → V0 is called the Banach adjoint of F and the Banach adjoint operator

∗ : B (V, W) → B W0, V0 is an isometry.

Finally, if V and W are reflexive, we note that F∗∗ = F, and the adjoint operator is an isomorphism.