Bases and Cones in Locally Convex Spaces

BASES AND CONES IN l,OCALLY CONVEX SPACES

FHA.NJ\ PAUL BOZEL, B.Sc., M.Sc.

A Thesis

Submitted to the Faculty of Grach.t~1te Studies

in Partial Fulfilment of the Requirements

for the D:::grec

Doc tm· of Philosopy

Mel-I.aster tin:i.ve_;·si ty May 1971 . DOCTOR OF PHILOSOPY (1971) McMASTER UNIVERSITY (Mathematics) Hamilton, Ontario.

TI'l'LE: Bases and Cones in Locally Convex Spaces

AUTHOR: Frank Paul Bozel, B.Sc. (Mel.faster University)

M.Sc. (McMa.ster University)

SUPERVISOR: Professor T. Husain

NUMBER OF PAGES: v, 73

SCOPE AND CONTEN'l'S:

The major results of this work include an isomorphism theorem for B-complete barrelled spaces with similar bases and a theorem which shows that the cone a.sf:cciated with a. cepnrating biorthogonal system in a perfect C.N.S. has a basis. We also obtain some applications of the former result in the case of dual generali:;:;ed bw:;es and some results concerning Schauder bases in countably barrelled spaces.

ii ACKNOWLEDGEMENTS

The author would like to thank his supervisor Dr. T. Husain

for the careful attention he has given this work and for his helpful suggestions and assistance in preparing this thesis.

The author would also like to heartily thank all those who were a source of encouragement to him, in particular Professor

H. L. Jackson and Mr. Shu-Bun Ng.

Behind each Ph.D. C'rndidate there sits a typist who has the sometimes exasperating task of reading the aspirant's writing, and so the author would like to express his appreciation to his typist

Mrs. Claudia McCarty.

Finally the author would lik<.:; t:o thank the Mdhster Mathematics

Department and the Ontario Government for the financial help he has received tr,:coughout his graduate c<:1.ree:c.

iii 'I'i\BLE OF CCNT_~;~Yl'S

IN'f'RODlJCTION 1

CI-L~PTEH I Preliminaries ~- 1 4 2 Locally convex sp2ces 3 MaPIJings and the dual 7 4 Inductive limits 9 5 Barrelled spacc;s 10 6 Duality theory. and rof1ex:Lve sp:-:ces lJ_ 7 Bases 13 8 Countably norr.c.cd opaces 20

CHAPTER II Isomorphism Theoreri1 and Applications 25 1 B-com;;lcteness 25 2 An isorJorphis:-:1 theorem 3 Inductive limits l~ Dual general:'._zed bases and further applications

CHJ\.P'F:CH III Counta'bly barre11ec!_ space::: and bases 43 1 Countably barrelled spaces 43 2 Permanence properties 46

3 The isomorph:i.srn -~heorem 4 Bases in locG.lly ccmvcx spaces

Normal cones ar:d bases in countably normed .spsces 1 Cones 2 r.!orual cones and bases

3 Eound0c1ly complete and type P bases, and sor::c exam~les

BIBLIOGHAPEY 72

iv TO MY WIFE MARGARET

v IN'l'RODUC'l'ION

The major results of this work fall into two main parts. 1'he

first part consists of an isomorphism theorem between B-complete

barrelled locally convex spaces relating generalized bases and simila1·ity,

and an analogous j_somorphism theorem concernine; countably barrelled

spaces and similar Schauder bases. We also give some applications of

these theorems and use them to generalize some known results to a larger

class of locally ccnvex spaces, and to inductive limit spaces.

'I'he second part consists of a theorem which gives sufficient

conditions for the cone associated with a biorthoeonal system to have

a basis in perfect complete countably normed .spaces. We also .show that

in perfect spaces every cone associated \vith a biorthogonal system where the functionals form a separ::1ting fam::i.l;y is normal ~and hence in

perfect complete countably normed spaces every c0c10 of this type has a.

basis. We e:1so obtain .some results concerning t!1e rele.tior.. between type

P and boundedly complete bas?ff ove:r a cone K.

The first chapter of this thesis is composed of tho1>0:: definitions and results from the theory of topological vector spaces, loca.lJ.y convex spaces and bcisis theory etc. which are needed in future chapters. In the introduction frequent use is made of the contents of Chc:tpter I withcut explicit reference.

In Chapter II, &ection l, we give some pn.:liminary results concerning closed gr<~ph theorems and then obtain an isomorphism theorem

( theorc11 2) for B-complete br.lrrel1.ed- spaces and similar gen:::ralized

1 2

Edwards (8), and was shown for the case of complete metric linear spaces.

It is worth noting that .Retherford and Jones (24) have shown that even in coMplete barrelled spaces the theorem does not hold, and hence our result is in some sense the best possible. This result is then used to advantage to generalize some theorems of Arsove and Edwards. In section

2 we apply the theorem again to the case of inductive limits, and in section 3 we further some results by Davis (15), dealing with dual generalized bases.

In Chapter Ill, sections 1 and 2, we give sc,me results concerni_ng countably barrel1ed spaces a.nd we show that the isomorphism theorem in

Chapter II holds for countably barrelled spaces and similar Schauder bases, thus extending a result by [fotherfo~d and Jones (24). It is worth noting that in (21~) the authors rer:iark that the isomorphism theorem concerning simjlar Schauder bases ie false if one of the spaces is not barrelled, however· our result holds for countably barrelled- S})ates. We also show that the weak basis theorem holds for countably barrelled spaces.

In section 3 we obtajn a. generalization of Gell:iaum's result (20) dealing uith the followinc question: If (x , f ) N is a Schnuder n n nc ba::ds in a linear topological space, can one give necessary and su:'.fid.ent ,--n conditions for (y ) N'' Yn L a.x. > (a.). M a sequence of scala1·s, n nc = _ _ 1 1 1 1C•• i=l to be a Schauder basis?

One of the main outstanding: problems in basis theory is: does every separE

In considering tl'.is problem, l;he idea of showing the existence of a basis for a cone has been studied by many people incJ.uding F'ullerton, Gurevic,

Gurarii, Singer, IJevin and McArthur, see for example, (23), (26), (27) and (28). In Chapter IV we show that in perfect locally convex spaces every cone of a certain type (i.e. associated with a separating biorthogonal system) is norm:i.l, and then we obtain the result that in perfect complete count<':l.bly normed spo.ces every sepnrating biorthogonal system is a basis for the associated ccne, and is in fact an unconditional be.sis. The result was previously known for a restricted class of Banach spaces (2.3), theorem 2. Finally He give some results relating type P and boundedly complete bases over a cone K, and an example to show that the theorem does not hold in complete countably normed spaces in general, and hence our result is in some sen~e the most general as y&t in this direction. CHAPi1ER I

Preliminaries

1. Terminolog;v:

Let X, Y be sets. We use the standard notation of x t: X, x i X,

x £ y' x ::: y' x n y' x u y and { ••• } • x x y shall denote

{ (x, y): x t: X, y t: Y}. The empty set shall be denoted by ¢ and

the zero of a vector space by Q. ft function or linear map between spaces

f: E --} span ScE E will b~ denoted F. The closed linear of a set - '

a topological vector space will be given by e.nd J.. J."' (x ) is [SJ' n nt:N a sequence the closed linear spo.n is given by lx J. The span of a sequence n or net (x ) will be given by sp {x,.. 1. Spaces will generally be a: a.cI ...... denoted by E, F, G, H. Given f: E --} F and H £ E, the restriction

of f to M in denoted .f I M. i'>'e denote inf:imuui and supremum by inf -1 If a map f is one-one we denote its inverse by f

2. Locally convex spr.;_ccs ----~------~

Definition: E is a topological vector space (in short a ·rvs) over ------· u a give-r.. field F' if E as a pointset is a t.opologic::il· spo.ce and a vector spo.c2 over F such that Hie maps

Cx, y) --7 x -1 y and (t, x) --) /..x are conti.:1.uous in :mth v::i.riables together, ' where x, y t: E, A. c F, and u is the topo],.ogy.

In the following we shall deal w:i th real spaces t~n1ess otherwis<~

.s'er;i "ied, an'.J aJ.l spacec: considered will be Hausdorff. 5

Definitions: Let E be a vector space.

(a) A subset M~E is circled if for each x c M, AX c M

f6i all j A! ~ 1, A. real.

(b) A subset A~ E absorbs a subset B ~E if there exists

ex. > 0 such that A B ~ A for all I A I.S ex., A -J O. A

subset B·.S E is absorbing if it absorbs each point of E.

M is called the circled hull of M.

(d) /, sub;;c;et H f;; E is convex if for any x, y c H,

A.x + (1-:\)y c H for 0 .S A. .S 1, /.. real.

'I'heorem 1: Let E be a TVS. Than there exists a fundamental system u LJ. of u-closed neighborl1oods of G such tlv1t:

(a) Each U c 'lj is circled and absorbing.

(b) For each U c tl there exists a V in L.t such that

v + v $: u. (c) u _ un c;ti . £93

Conversely, if F~ iEi ct vector spci.ce and ~ a filter .base satisfying (ri.), (b) and Cc), then there- exists a Hau·sdorff topology u on E such that E is ci. TVS and is a fundamental system of u J. neighborhoods of

-·-Proof: A proof may be found in (111), (Chapter 2, Page 14).

A TVS E is locally convex if there exists a u fundamental system of convex neighborhoods of 9.

(b) J.,et E be a real vector space. A semi-norm p on E

is a re<•l valued function. p: E -7 R (R = reals) such thrit 6

p(A.x) = JA.\ p(x) for all A. c R, x c E and p(x + y) ~ p(x) + p(y).

We observe that p(Q) = 0 and p(x) .2: 0 for each x c E.

If p(x) = 0 implies x = g then p is called a norm.

Theorem 2: Let p be a nonempty family of semi·-norms on a vector space E. For p c P, define - V(p) - [x e E·. p(x) < i} . Let Lf be the family of all finite intersections

and for all 1 ::; k ::; n. Then there is a unique topology u on E such that E is a locally convex TVS and 'LI is a fundamental u system of convex neif,hborhoods of the origin.

Proof: A proof may be found in (29), Page 146.

Prot:osition l: (a) Let E be a locally convex TVS, and M Si,,E u u a u be of (qA.\c ·• subspace. Let given by the family semi-normG 1 Then the quotient space E/M and M are locally convex spaces with the topologies defined by the .farrily of sem5 -norms CiiA. \cL, (where

inf and respectiVely. xcx.

(b) Let (Ea.)acI be a family of locally convex spaces. Then the product re E i.5 also a locally convex 'l'VS with the product topo1ogy. cicI a ----Proof: A proof may be found in C13)(Chapter 2, g 5).

Definitions: (n) A locally convex TVS }~ is complete if every Cauchy u net converges. 7

(b) A subset BCE is bounded if it is absorbed by every - u neighborhood of the origin.

Theorem 3: Let E be a locally convex TVS, and MCE a closed u - u

linear subspace. Sup~ose there exists x c E , x I M. Then there u exists an f: E ~ R such that f(y) = 0 for y c M and f(x) 1, u = f a continuous linear G.• e.• f(A.x + uy) = A.f(x) + uf(y)) map.

---Proof: A proof can be found in (29), (Chapter 1, _,§ 7, Page_ l.j2) 4

Proposi ti9£~: Let E be a locally convex TVS defirn:-d by u (qA.\cL"

A subset BCE is bound.::u if and 0~1ly if every is bounded on - u qA. B.

Proof: See (13), (Chapter 2, £ 6).

Definiti~: (a) Two fam-ilies of semi-norr.is on a vecto!' space E

are equivalent if they define the same locally convex

topology on E.

(b) A family c;}: of semi-norms on a vector space is eaturated

if lof any finite subfamily (q.). J of ~ , the semi-norm 1 J.C q(x) = max q .(x) is also in 1 ~.c.J Remark: If (qA\cL is a· family of sewi-:::-.orms defining a locally convex

topology on a vector space E, then the family obtained by taking, for

every finite subfamily (q.). J' the mnx q., is an equivalent sa.turated 1 1£ icJ 1 family of semi-norms.

Definitions: (a) I.et E, F be two vector spaces. The set of all ---~- ...--,- linear maps f: E --> F is a vector space with poinh:ise

addition and scalar mul tipl:ic::i tion. 8

(b) E# ; the set of all linear mnps f: E · -4 R is called

the algebraic dual. If E is a TVS, the set of all u continuous linear maps _f: E --1 R is called the topological u

1 dual E_' • , Cl.early ...."'' :J cE- #- • The elements of E#are

called linear functionals, and those of E' are called

continuous linear functionals.

Pro12ositicn 3; Let be TVS's. A linear map f: E ---i F u v is continuous if it is continuous at the origin-.

Proof: See (30), (Chapter 10, S 5, '!'heore~ 1).

Definitions: Let X , Y be topologfoal spaces. u v

(a) A map f: X ~ Y is a.ln1ost continuous at x c X if u v for each v-nei8hborhood V of f(x), f-1(V) contains a

u-neighborhood of x. f is almost ccn.tinuous on X if

it is so at each point.

(b) x ~ y A me:.p f: u . v is almost op~n if, for each u-neighborhood U of x c X , f(U) contains a v-neighborhood of f(x) u (c) Given a map f: X --> Y , the sub:;et \ex, f(x)): x c u v t of X X Y is called the grapl-: of f.

(d) Given two TVS's a map f: is an ""u'"' - F' v

isomorphism if f is linear, one-one, onto, continuOl~s

and open.

Rerr!_~1=.:~: -All isomorphisms will be on to in the sequel unless o:h0rwise indicated. An isomoq)hism which ia not onto is c.rnlled an iso:norphis:n into.

IJet E be a locally convex TVS whore u is given by the u 9

saturated family of semi-norms, and F a locally convex 11VS, v v given by A linear map f: E -.-:,· F is continuous if u v and only if for every there exists a and M>O such that

for all x c E • u

Proof: See (13), (Chapter 2, ~ 5 page 97).

.r • Remark: If a linear map .L • is continuous, then it is uniformly continuous.

Proposition 5: Let E be a TVS arid F a cor.iplete TVS and A ~E u v u a subset. If f: A -.,l F is a un~formly continuous maD then there v - ' exists a uniformly continuous extension of f to A.

Proof: See (13), (Chapter 2, g9, Proposition 5).

4. Inductive limits

Definitions: (a) Let (E ) be a family of locally tonvex space~, a. ex.cl . E a vector space and (f ) I linear maps f : E ~E a. ()'.£ O'. a. for each a. c Suppose f (E ) • 1'he inductive r. U a. a. a.er limit cf (:S ) I is E where u is the finest locally a. o.c u convex topology of E such that each f is continuous. a. (b) If I= N (the natural numbers), ·then E is called the u generaliz'd strict inductive limit of the

i.e. F; is the un~on of a stric;:tly increasing sequence

of its' subspaces, and if the inductive limit topology on

E induces the same topology as that of E , then E is n u called the strict in1uctive limit of (E ) r~ • n nc ~. 10

Remark: If E is the strict inductive limit of locally convex spaces u (E ) N' then E is Hausdorff if each F; is. nm:: u n

Proposition 6: Let E be the strict inductive limit of (E ). N• u n nc Then if each E is complete, so is E • n u

Proof: See (2), (Chapter 2, S 2, Exercise 9).

2. Barrelled STJaces

Definitions: l.a1' Let E be a locally convex TVS. A closed, convex, ' u

circled, absorbing subset Bs,E is called a· barr~l. u

(b) E is called a barrell,~d sp~:.ce (sometimes t.,.;space from u the French espace tonnele) if every barrel in E is a u neighborhood of the origir1•.

(d) A topologicRl space x is a Baire spr:tce if it can't be u

expressed as the cour: fr~ble union of nowhet'e dense sets.

Prot:osition ?: A Baire locally convex TVS E is barrelled. u

. Proof: See (13), (Chapter 3, ~ 6, Proposition 3).

Remarks: (a) If E is a barrelled TVS and M c: E a subspace, --- u - u then the quotient space E/H is barrelled.

(b) If (E) I ·is a family of barrelled TVS's, then the ex. a.e:: product with the product topology is bvrrelled.

Cc) A closed subspace of a, bRrrelled space is not neces[;arily

bat>relled. J . 11

(d) Inductive limits of barrelled spaces are barrelled~

/ (e) Every Baire space hence F'rechet space (complete metric

locally convex TVS) is barrelled.

(f) A barrelled space is neither necessarily metrizable nor

complete.

6. Duality theory and reflexive spaces

Let E be a locally convex TVS. We present here some topologies u on E and E1 which play an important role.

Definitions: (a) The coarsest locally convex topology on E such

that the map X -7 f(x) for each f c E' is continuous,

is called the weak topology, a- {E, F~') on E.

1 (b) The weak* topology, o- (r~ , 'S) is defined as the

coarsest locally convex topolog~r on E' such that the map f-~· f(x) for each x c E, ·is continuous.

Remark: c-(E', 1'~) is precisely the topology of pointwise convergence ") .p. which is indu~ed from the product to:;::iology on RE = (r 1-1here J. • E -> R J•

Definition: Let E , F be two TVS's, not necessarily Hausdorff. u v Cons1der the vector space ;J:,,~E, F) of all continuous linear maps f: E --) F with pointw:ise addition and scalar multiplication. Let u v G be a chrns of subsets of Eu· One can defjne a topology of uniform convergence over sets M c G as follows: Let V be a neighborhood of the origin in ,'ti' and Let T(M, V) be all continuous v linear maps f c ;((E, F) such that f(H) s; v. The collection of all

'l'( M, V) t M c G V a neighborhood cf the origin, forms a subbase for

a topology on d'.'CE, F) called the b -·topology. See (2) 1 (Chapter 3, ~ 3). 12

Remarks: (a) £er;, F) with an · b-topology is compatible with the

vector space structure if and only if f(M) is bounded

in F for.each M and f ;/!.(E, F) • v cb c (b) If F is locally convex, then the G -topology is also. v (c) If c_:; consists of bounded sets of E such that u UM McG is total (the set of all finite linear combinations dense)

in E and F is locally convex, the!'l the ~ -topology u v is a Hausdorff locally convex topology.

Definitions: (a) Let· E be a locally co~vex 'l'VS. Then the strong u topology j:> on E' is the (:;-topology where G is

the collection of all u-bounded sets of Eu· is called the strong dual of Eu • (b) The -top0logy on a J,ocally convex TVS E where G G u consists of all convex, circled,

sets of E' is called the Mackey topology, denoted by

"C(E, E').

A barrelled TVS is a Mackey space.

Theorem 4: (Mackey) Let E be a locally convex TVS. 'l'hen a subnet u M ~ E is o-(E, E') - bounded if and only if M is t.(E, E') bounded. u

Proof: A proof mny be found in (14) (Chapter 2, ~ 9, Theorem 8).

Definitions: (a) The dual of is cfll.led the b:i.dual of

(b) If E,P, =E (aleebraical1y) then E is called semi-reflexive. u (c) If E'~'~ :;::: E {topologically) then E is called reflexive. tl .13

Clearly a locally convex 'l'VS E is semi-reflexive if it is u reflexive.

Proposition 8: Every barrelled TVS E is a Mackey space. u

r. 9 ---Proof: See (14), (Chapter 2, e ' Proposition 16).

Proposition 9: A reflexive locally convex TVS is barrelled.

Proof: See (14), (Chapter 2, § 9).

Proposit~on 10: The strong dual E'~ of a reflexive TVS is also reflexive.

Proof: See (13), (Chapter 3, $ 8, Proposition 7).

Definitions: Let E be a TVS, u (a) A biorthogonal system (x , f ) I in E is a family a. a. m:: u

T of points in E (not necessarily countable) and (xex. ) a.c_

a family (f ) f c E' for all 0: c I, such that a. a.cl' ex.

f ( J' -· s i.e. $ 0 if a/. ~, 1 if a. µ. a ,XIJ a(3 • Ct,~ = - (b) A cour.table family (x ) in B is a basis if for each n nc:N u x c E there exists a unique sequence of scalars (a ) u n m:N co n such that x = a x i.e. :x = lim a.x. in the I n n n4 CD L 1 1 1 i=l

topology u. Hence [x ] E Given x n = u" =

can define f by f (x) a • This gives a unique family n n = n of functionals called coefficient or coordinate functionals,

and f (x ) 8 l'/11en all t.he (f ) are contim.wus, n m = nm· n ncN , .r we ~all (x ) o:..~ .1 ) a Schauder basis. n rn:N ex n ' n m:N Remarks: (a) 'l'he above definition of b3.sis was introduced by Schauder

(5) with E taken as a Banach space, and has subsequently u been extended to TVS's in general.

(b) The coef.ficient functionals are not always continuous as

the following example shows. Consider Ero the space of

all real sequences of finite support, with norm

n = 1,2, ••• Jwhere 1 Define (x ) N by e where n nc = el' xn = el + ~ n h+I' We claim (x ) . 'is a basis, en = (O, o, ••.' 1, o, n ncr·J

since for a (a ) . £ E CD suppose a 0 for n = n ncN = > r. ' r n Let the coefficients be k t k for tl = a1 -L. ak, k = ~ k=2 r r k 2. Then a since t x o, o· ') 2: \xk = [_ k k'= (tl, ... k k=l '

1 \ -· t , o, ~ ~ ) r r .

a , C, ••• ) =a. Also, the are unique, .-.' r

since if b 0 for n > r, and n = r 1 a b • By :repeaUng thi.s \lfe get b =a. - b a - f__ n = - n n L 1 = 1 1 .l k=i K k:.ol 1'he coefficient given by f Ca) = t ... 1 1 is not continuous. T!1is is seen by ob.serving that xn--) x 1 as r.. -7 cn, and since r (:< ) = 0 for n = 2, 3, ••• , Hhile 1 n . f ) 1, f cannot be continuous. 1 Cx1 = 1 (c) A TVS with a (Schauder) basis is separable.

The following theorem due to Newns shows the.t for a very large

class of' sp:::..ces, th: (!Oe:fficient funct:1.onals are cont:i.nuouc;. 15

'l'heorem 5 (Nei":rw): Let E be a complete metric TVS over the real u or complex field. Then every ba.:.;is for is a Schauder basis.

See (20), (Cha.pter 9, .£ 4, 'l'heorem 2)

The question of existence of a Schauder basis in Banach spaces

is yet unsolved although many people ho.ve ma.de an effort in this direction.

_The following shows that the important question of existence canr10·t> be

answered generally in the affirmative in a general TVS.

Example: Let S be a compact intervci.l of the :reals. Let L (S) be p

the set of all equival:ence classes of measurable functions f: S ~ R,

Ircx)I Pax < + ro. It is knmm (20) (Chapter 9, s 5,

Theorem 2) th9.t L (S) with 'I is a complete quasi-normed TVS. p II I.

(i.e. 11 f\I = o if and onJ.y if f = O, II f + glj _:;\.Ir\\ + \\gll ') . and L (S), 0 < p < 1, provides examp1es of completP quasi-normed spc:.ces p which have no Schauder basis, since L (s)' = Io1 This example is p l I • M. M. Day, see (20), Chapter 9, Corollary 3). However L (S), p I )l with .-: (J If(x) j p P 1 < p

Definitions: (a) A biorthogonal system ( f '1 in a TVS E xa.' a. acI u is called separatinlj if f (x) ~ 0 for al1 a. ~ c I gives x = g• One can also refer to

being separating.

(b) A biorthogonal system (x , f ) I in a TVS E is a a. u ac u generalized basis if it is separating. If in addition to

this, the (x ) I is total i.e. b: J -- E , then th(; a. a.c ~ u system is called a Msrkuschevich basis. 16

, (c) A biorthogona.l system is called maximal if it is not

contained strictly in any larger biorthor;onal system.

Proposition 11: Let (x , f ) N be a biorthogonal system in a 'I'VS, n n nc Eu. Then (xn' fn)ncN a Schauder ba.sia implies Markuschevich basis

implies generaJ.ized basis implies maximal biorthoCTonal system.

Proof: The implica-Lions Schauder to Il,arkuschevich to generalized basic.

follow directly from the definition. T'nat a generalized h.asis is a

maximal ·biorthogonal system follows easily. If (x , f ) N were not n n nc

maximal, then there would exist a point y £ y /. g and g c: E' such Eu '

that g(y) ::::; 1 and f (y) 0 for all n £ N. This is a contradiction n -

since f (y) ::::; 0 for all n gives y Q, and g c E' gives g(G) Q. E .• D. n ·-- = o.

The implications .::u·2 not reversible &s the following examples

show.

~ Examnles: (a) Let E be the Frechet space of all functions analytic u ori the open unit disc ( Iz I< 1) topolog;i~:;ed by the metric of uniform convergence on compact sets. Choose ''a" any complex number such that

0 < 1. Set x (z) ·- (z a)n for n ::::; o, 1, As (7) shows, < la! n -

(x ) form8 a Markuschcvich basis in ~J and the corresponding coefficient n rn::N "'u functionals are given by o-(n)(a.) f ( g) = .l;L_____ n n! n = O, 1, •••

(b) Let I be a countably infinite set with the discrete topology and E the space of all bo:mded rr:al valued functions on I. This is a u

Banach space with sup norm. For each ex. [, I, defin,;d x by x fo) 1 ex. a. - and 0 elsewhel'e. The point functionals f (x) = x(a.) are all continuous a. and f ) forms a biorthogonal system. For x c E if f (x) 0 (xa. ' a. o:cI u ex. = for all a. then x Qt hence (x f ) is a generalized basis. - a.' a. a.er E ·is not separable (see (29), Chapter 2, Page hence (x ) u 3, s 89)' a. a:.i::I cannot be total, so the system is not a Marirnschevich be.sis.

2 n x (z ) 1 + z + z + ••• + z n = (n)(O) IS. - JL_(n+l)(O)__ and (f ) by f (g) = for n ncN n n! ·· (n+l) ! n O, 1, and g c E. Trivially (x , f ) u forms a biorthogonal = n n ncn system, but (f ) N is not separating. To see this we use the foe t that n nc f (g) - 0 for all n c N if and onlJ' if g is of the form g(z) n = 2 c(l + z + z + ••• + Zn + • • •)' wher~ c is ar; :trbitrary scalar. Hence

(x , f ) -,,r is not a generalized basis, but is maximal s5_nce if not there n n ncl'I exists an h c E' such that h(x ) 0 for all n but h I 0 the zero n = functional. This is impossible sinGe every h c E' has the representation

Q) '(n) (O) h(y). == I:. x_._,--- h , h suitably chosen complex numbers. Hence n=O n. n n (x f ) is a ma:;dmal biorthogonal svstem. n' n rn::N " The follm1jng results affirm the exL3tence of tho o.bove types of bases.

Tlt<22.!:_~.£..J1·hrkuschovich): There exists a Narkuschevich basis for every scpa;:-able Banach spnce. -.-·Proof: See (20), (Chapter 9, 81, Page 116).

Theorem 7: Every separable Hilbert space (complete inner-product space) has a Schauder basis.

Proof: See (20), (Chapter 6, .£ 1, 'l"neorem 1).

As mentioned before the problem is yet unsolved for Banach spaces·, however the following is known.

Theorem 8: Each infinite dimensional Banach sp:;ce E contains an u infinite dimensional subspace with a basis.

---Proof: A proof may be found in (20), (Chapter 3, ~ 3, Theorem 7).

Definition_: Let f ) be a generalized basis in a complex or (xa. ' a. w::I real TVS Let RI be the linear space of all indexed families Eu • (a..) .•, the operations being pointwise relative to the scalar field. J. J.C.L.

Define by :r._ (x) == (f (x)) is called the coefficient 1' a. a.c r· ~ map determined by the CrQ. )O'.C r·

Remarks: (a) is linear since

(ax + by) = (f (ax+ by)) T = (a f (x) +bf (y)) I a. • a,;:;__ a. a. O'.C

=a(f(x)) T+b(f(y)) I a. a.c_ a. a.c

= a p(x) + b ~ (y).

(b) 'Ihe property of being a separating family of functionals is equivalent to 4 being a one-on_e.mo.p.

We now pre.sent some further fa.cts flbout coefficient functionals and generalized bases~ - 19

A biorthogonal system (x , f ) I in a TVS E is a a. a. a~ u generalized basis for E if and only 5.f is one-one. u p

Proof: The proof is evident from the definitions.

Theorem 10: A given family ( f ) e: }-:;• can be the family of coefficient _ a. ac.I

functionals for at most one generalized basis in Eu·

Proof: If there is another generalized basis (x', f ) I for E , ex. a. a.c u then fc/xp = 8 = fo:.(xµ) for a.' p c I. Hence fa.(x~ - xP) a.f3 . - fa. Cx fa.(xf3) :::; 0 for all a c I, so x' -- Q.E.D. 1p - f3 xf3 •

A generalized basis (x ) - for a locally convex 'l'VS E a. acJ.. u has a unique family of coefficient functionals if and only if (x ) a m:I is total in E • u

Proof: Let (x , f ) I be a total ~eneralized basis in E such that a. o:. ac. u (g ) I is another distinct family of coefficient functionals. Then o: a.c

f (x,) = g (x..) = 8 r:< for o:., ~ c I. Hence (f -g )(x) = 0 for a. c I, a ~ a. ~ a.~ a a. x c sp(x }. Since (x ) - I iD total an:l f - 5 is continuous, o:.c Ia. {.(. 0:.£ 0:. a.

(f )(x) for all x c E· that is, f ::;. g for all (j, £ a ga. = 0 u' a. ·a. I.

Conversely, suppose that ( f ) I is unique, but (x ) is a ac a. cu::I not total. Hence by 'l'heorem 3, there exists an f c E 1 such tlmt

f(x ) == 0 for u. c I, and f /- o. Let i3 c I, p fixed. 'Then observe a.

H::>nce- (x , f + On'\" f' 1· .;~-'-'· s ., ~ a. a.~ ac biorthogonal system. To see that the system is separating, assume

(f + 8 f) (x) 0 for <.."t c I, and some x c Then f (x) :::. a a.S - Eu • a. -f (x) fa.(xµ) := f (-f(x) x ) for a. c r. H-;nce (x) = C-f(x) xf:S) • Cf.. µ ~ p 20

But since (xa.)o:.cI is a generalized basis, p is one-one, so x -f(x) x~. Therefore f (x) = f (-f(x) x ) = -f(x) f (xA) = 0 for = a. a JJ - a.I"' a. c I because f(x) = 0 for all x c sp{xa.}. Since (f,t)a.cI is separating x = 9. So there exists another generalized basis

(x , f + i' f) Contradiction! Hence (x ) is total. Q.E.D. a. a. 0 a.P a.cl" a. a.er

Remark: By the label "weak" as applied to any of the types of bases previously con:>idered, we shall mean that the topology on the linear space being considered is the weak topology.

Theorem 12: E~vcry weak Schauder basis in a barrelled TVS is a

Schauder basis in the initial topology.

Proof: See (1) (Theorem 11).

Let E be a locally convex TVS. Then eve:-y v:eak u Markuschevich or generalized oasis in E is one for the j_ni tial u topology.

Proof: See ( 1) , ( TheoreI1 10).

8. Countably normed space.~

I.et 11 j j , lj !1 , • • • be a countable system .of norms definc:d 1 2

Using these norms we can introduce A to_r>ology on E as follows. I.et p be an arbitrary positive integer and c > o.

Define neighborhoCl\ls of g c E by

up,c (Q) = ~ x c E: Hx ti 1· <· £' 1 ;$ i ~ p} .

We ob.~~crve that the following propcl"ties hold:

(a) 0 c u for all f;. p,c p' 21

(b) Given U and U defined as above, there pl,cl· p2,c2 exists a third neighborhood contained in u n u i.e. plr:l p2,c2 c u u n u where u is such that (; min {c , c2} p,c pl,£1 p2,c2 p,c = 1 and p = max lPl' P2} •

(d) Given a neighborhood U· of g, there exists a neighborhood

W of g such that w+ ws u. i.e. If u = [x £ E: II xi Ii < c 1 :5 i .S p} one can take w::: [ x c E: HxHi <%,1.:Si~p} • This follows directly from the triangular inequnlity.

(e) Given x (; u there exists a neighborhood v of G 0 p,c such that + 5:- SimvJ.y take max '•~'e note XO v up,c· II xoJl i = 1\ i:,Sp that '<.~ c, so one can take v ::: £ E: Ix < c-Y\ 1 :5. i. :5 pJ I\ tx 1.1

(f) Given a neighborhood u and I 0 a scalo.r, there p, (: a. exists a neiGhborhood W of Y such that a.\~ c::: U -· p,c 1 :5 i :5 pJ does the trick.

- . (g) Given a neighborhood U and X c E, there exists p,c 0 an a. JSUCll that cS'xo c u for < a. .. Let = a. • j 1, 2, ... p,c 1°1 II XO Il j J then put (; a. = ~;(a.]--:-:-:·:-;~). . p (h) For any neighborhood v of g there exists nn c > 0 ,.. such thnt 8v ...... -· v for \81 < £. To see this we simply take c = 1 •

By Theorem 1, this family of all u defines a topology u p,c on E such that E is a 1'VS. and the \U l ,., , is a nei,0:hborhood u ' t p,c J pc1~;c__,0 basis of the origin. 22

Definitions: (a) Two nor11:s E II [j 2 on a linear space are COr.lp&rable if there is a c > 0. such tho. t II x l.11 .s c nx Il 2 for all x c E.

(b) I II I\ !12 on E are compatible if every sequence (x ) N which is cauchy with respect to both norms and n nc _converges to zero with respect to one 6f them also converges

to zero with respect to the other. A countable family

of norms is compatible if any two are compatible with

each other.

Remark: In the case where one of (E, 11 11 ) = E and == E 1 1 2 is not complete, they can be completed in the usual way. is the stronger norm i.e. 0) then one can establish a map of E into E where every elern211t x c E 2 1 2 is defined by a sequence (x ) "c:: E which is cauchy with respect to n nci~ '""' 11 11 , and hence \·J:~th !! / j . and so de.fin.es a uniquely determined 2 1 element y c E • To insure that this map is one-one we require the no2·ms 1 ' to be compatible, Hence if there are two comparable and compatihle norms

E and x f 1 1 x x c E, · the 11 11 1 , 11 1'2 on · [I j .S Cl I!2 for then completions of E ~ E with respect to j I j 1 , 1 2 1 to have the following relationship. /\ A E ::> E ::::; E. We note that under this identification evr,,ry 1 2 element x c E is carried into itself.

Definition: A countably normed space E is a locally comrex TVS . ·- u where the~ topology can be given as described previously.

---·Remark: The defining ··neighborhood system of the orig.in cc:i.n be counto.ble 23

1 1 1 by to.king t: :::; 1, 2' 7.' ... ' n ... and hence the first axiom of :J countability is satisfied. Also, one cnn always con.sider the sequence

of norms to be nond·ecreasing. ~ i.e. II xl! 1 lrx/j p ~ • • • for x c E, by taking an equivalent family give by llxll ~ - 1~~p{H xii i) Definition: A sequenqe in a countably normed space is

cauchy if it i~ cauchy with respect to each norm.

One can consider the completion of each ( E., N say II H.J. )J.C .

E. and we have that El:> E2 ::> ... ~ E .::> :::> 1!~,, where E. is the J. p ...... 1 completion of E with respect to II I! i. CD

The.space E is complete if 1-1.;.1d cnly if E == E ;.1here u n p p==l E fa the completion of E with re.spec t p to 11 If p •

Proof: See (9) 'Ch, .apt·er 1, _(,'_ 3).

In what follows we shall ah:ays d~al with complete countably normed spaces, denoted C.N.S. unless otherwise indicated. We now give some nontrivictl examples of' such spaces.

Let K(a). be the set of all infinitely differentiable functions on (- CD' m), 'trhich "lanis11' outsids \ x \~a, with norms - (p) I] given b:r ii rfI == max {I f(x) I , l f I (x) , ••• , I_ f (x) I , p lxL~ a f6r p == O, 1, 2, ••• K(a) is a C.N.s., (9), (Chapter 1).

Let Z(D) be all analytic functions f in the region

D ::: { z : < a} wher-e a is a posit:i.ve rea.l number w:i.th the no1·ms

(' I • 1· given by ==max , z,I .ap) ) where (a ) F is a monotone l!rllp tlrCz) 1 ~ P pc.• increasing sequence such that lim a -· a, Z(D) is also a C.N.S. P-~CO p See (9), {Chapter· 1). Definition: A countably Hilbert space (C.H.s.) E is a countably u normed space where the topology is_ definr:d by a countable system of scalar products ( < >) N which are compatible (the norms -, p pc ll xl! p = \/ p x c E, p -- o, 1, 2, • • • are compatible). The topology is defined in the manner described previously.

We note that every Banach -(Hilbert) space is a C.N.S. (C.H.S.) respectively~

Theorem 14: Let E be a complete C.H.s. Then E is reflexive. u u

Proof: Sr;e (10 ) (Chapter 1, ,.;c 3,. Pace 61). CHAP'l'ER II

~-B-Completeness

TVS E ·Definit:i.ons: A locally convex ·u i.s R--cor.inlete if any linear, continuous and almost open map of onto s.n.y

locally convex TVS F is open v (b) A locally ccnvex TVS E is B -comnlete if any linear tl r . continuous almcst open one-cne map of E onto any u locally convex TVS F is open. v

Proposition l; Every B-comolete space is Br-complete.

----Proof: Tne proof folloHE; irnmediately frcm the definition. .I Every Frechet spacr1. iG B·-complete.

----Proof: S•.Vi (14) (Chapte1· 3, $ 2, Tneo:rern 2).

Each Br -·complete . (hence also B-complete) locally convex TVS E is.com9lete. u

---·Proof: A prcof ma.y be fo:m.5. in (14) (Chapter 5, ~ 3, Propo.sition 10).

Let E be a B-cornplete TVS and. H c. E a closed u - u subspace. 'l'hen M is B-complete.

Proor:: See (11+) (Chapter l~,

I.,et E F be two locD-lly convex .spacl=:s. Let u~ v f: Eu-·--> Fv be a linears continu:>us, almost open onto rr.ap. If !""! u is B~complete, sc is Fv· Proof: See (11~) (Chapter 4, g 1, Proposition h).

Corollary 1: Let E be a B-complete TVS and M s;.E a closed u u subspace. 'I"nen E/M with the quotient topology is B-complete•.

Proof: Se~ (14) (Chnpter 4, S 1, ?reposition 5).

Eropositt~.n 6: Let E be a l''recht't space. Then the topological dual u

E' is B-complete for any topology v such that ""t:(E', E) ::J v ::> c where i:(E', E). is the M3ckey ·topology and c the ~-topology of all u-compuct subsets.

Proof: See (ll~) (Cha.pter li-, S 1, Proposition 7).

Definitio~: Let X , Y be two topoloeical spaces and f: X -> Y u v u v l a The graph of f is f(x)): x c x and is a subset map. { (x, u J , of y The graph of f is closed if it is a closed subset of xu x v . X X Y with the product topology. u v

Let Xu' Yv be topologfonl spaces. Then f: X --7 Y is u v closed if and only if for any net (x ) c X with x --·7 x ar..d a acI ~ u a f(x ) -~ y, it follmrs that f(x) = y. a

Proof: See for example (30), (Chapter 11, ~ 1, '11heorem J.).

Theorem (Hobert.son ar.d Robertson). Le:t E be a B-cor.Jplete TVS l; ·u arid F a barrelled TVS. Then if g: F ..._.; E is a linear map with v V. ~ closed graph, then g is continuous.

Proof: See (J_lf) (ChA-pt€r lt, ,~ 5, Theorem 8(6)).

Let -'u'F. .ii' ,. be B-ccmplete b9.rrelled spaces. 'I'hen (a) any linez.'.:r' nnp f: E ---·'>Ii' with u v 27

closed graph is continuous

(b) any continuous li_near mapping f: E ----} }' onto is open.

Proof: The proof of (ci.) follows trivially from the theorem. Q.E.D.

Definitions: Let E F be TVS's. Then u' v (a) two Schauder bases ( x ) . c: E - and ( Y ) S F n nc1'1 - u n rn::N v are if any sequence of scalars siiitilar for (an ) ns N' CD 00 & x converges if and only if a y converges. n n r_ n n- n=lr n=l

(b) Two generalized bases (y ) c :B' ex. asI - v are similar if the cor:._·csponding coefficient functionals

such that

l. (E ) = "t' (F' ) '±' u v

Proposition 7: Let E be a linear space with u, v two topologies on E such that E are TVS's. As~;-:_1ff·e th.~::-e exi·3ts a separating Eu ' v family (f ) of real valued functions suc11- that each f is continuous a. a.cl a. with res:.Y-~C t to u ai:d v. Thc-:1 tL: identity map i: E --} E hc1s u v a clo3ed graph.

Proof: Let (x.J(.) - be a net in E such tbit converges to --- P pc.J u (xp) ;3t:

-P. ( , .. ) ~, with the USUE:1.l tcr)o1ot;;l s_re unique, f (x) for ci.J..J... a. [. -'-. a. = J..c:x. '•/ 28

Since (f ) I is a separating family of functions, f (x-y) ::: 0 for a.~ a. all a c I, implies x-y = G or x ::: y. Hence the graph is closed.

Theorem 2: Let E and F be a B-complete barrelled spaces. Let u v (x ) I be a generalized basis in If T is an isomorphism a. a.c Eu • T: E ---:} F and Tx =· y for all a. c I, then (y ) is a u v ('' ex. a. a.cl generalized basis in F and (x ) I is similar to (y ) 1· v q. 0:.£: . a. CX.£: Conversely if (y ) is a generalized basis in F similar to ex. a.£: I v (x ) ; then there exists an isomorphism T: E --~ F such that a. CX.£: 1 u v y Tx for a. c I. ex. = ex.

Proof: Assume T is an isomorphism of. Eu onto F • Let (x , f ) I v a. am: be a generalized basis in E • To show that (Tx ) . is a generalized ·· u Ct.. a.£:! basis we need to show that there exists a family (g ) ~ F' such a. acI that (Tx g ) is a biorthogonal family and ( g ) is separating. a.' a. a.cl a. o:£: I

Define g ::: f T-l Th 0 n the g 's are continuous (ga.\u:I by a. ex. 0 • . ~ ex. -1 1 and linear since f ahd. T are and g (Txp,) = (f 0 T- ) ('l'xr~) = C'.. ·ex. /' a. I-' fa. (xf:l) ::: 0 so (Tx ) is biorthogonal. Also if gcx.(y) - 0 a.~i' c' ' ga. a.cI T for all a. c .... , then (f T-l)(y) ;:: 0 for all a. c I, and hence ., a. ~ T-_1..(y) Q. Since T is ·;ci.r.. isomorphism, y ;:: Q in Also, (x f ) = Fv • a.' a. a.£:I is·similar to (Tx g) since "-r"(y) = (g (y)) = (g ('l'x)) _ = a' a a.cl a. a.cI u a£:l ( f (x)) I ::: . X (x). Ct.. a.£: 'f

For the converse, assume (y ) is a P"eneralized basis in a. ca: I 0 F similar to (x ) in E • Then J: (E ) = "-YO' ) = Ji- say, v a. a.£:1 u '±' u v whe1'e ~ , "t.1 are the corresponding coefficient maps. J+ is clearly 1 a subspace of R under pointwise addition and scalar multiplication, and A- is algeb;:'aicEt1ly ~sor.iorph:i..c t(1 E and F respectively. Hen,.:;e we can induce the topologies u e..nd v on J-t as follows •. We define· 29

1 A ~ to be open if and only if ~ -l(A), y- (A) are open in E v4 u and F re;:>pectively. Hence and are B-complete barrelled v Ji-u flr v J_ spaces. Now define T by T = ·y- 0 f. . Clearly T is a linear mapping from E to F and is one-0!1e and onto since "1/ and p are.

In Proposition 7, take E ==A and (h ) I defined by h (z) = f (x) ::. a. o:.c 0:. a. g (y) where z ,+, (x) (y) o . a. = ':t' = Y (ha.\:.c1 is a separatine; f3:mily of linear functionals because (f ) ex. ex.er and (gcx.)a.cI a..~e and ha.(z) = 0 implies f (x) = g (y) = 0 for ((. c I, where z ::: (x) ='-VCy). ex. (J, . 1 Hence x and y are Q in E , F respectively, so z Q because u v = and are one-one. The h are continuous with rei:;pect to u p Y a. and v because f and are continuous with respect to u and v ex. gcx. respectively. By Proposition the identity i: v4- --)Ji~ has 7, u v closed graph and by Corollary 1 of this chapL~r, u is finer than v and hence u = v. T is the desired isomorphism. That Tx = y for ex. 0:. all a. c I follows by similarity since for a. fixed but arbitrary, - (''J-1 p )( )- •"\.1..J-l( ( )) ·- "\..1/-l(O' ( )) - Tx - ~ 0 X - T fA X 0 I - 1 u~ Y ~ I - Y • Q.E.D. 0:. 0:. . f-' Ct pc · f-', 0, f-'C 0:.

Remark: Theorem 2 extends a result due to .:\.rsove for metric complete linear spaces.

This result allows one to obtain the folJ ow:ine, \vhich \vas known only for c,~mplete metric linear spaces. (see (1) Theorem L~).

Let E and F be B-complete barrelled spaces with u v

. (x ) (y ) t6tal generali~ed bases in E and Let I'~ I a. Ciel' a o:.cI u Fv • , F' such that I - I' is finite. Define F'l' · 1 to be the closed linear spans of with the relative topologies of E and (xa ) o:.c I', u :B' respectiv01y. If ·r_ is an isomorphism T : E -} :F' such that v J. 1 1 1 30

T xa. =ya.' a. c I', then T can be extended to an isomorphism 1 1

'I': E ~· F such that Tx: y for all a. c r. ex. = a.

Proof: Observe th2t by the definition of E and F , 1 1 (xa. ) a.c I'' (y ) are total generalized bases, and this follows by defining o: a.c I' f' = f E and g ' g where ( f ) I and (g ) are the 0: 0: r 1 ex. = a. IF1' 0: o:c o: a.er corresponding coefficient functionals. Since T is an isomorphism and 1 T x ::. v for a.cI', by Theorem 2, (x ) I' is similar to (y ) 1 ex. Va. a ac a o.!: I' • Observe that E = E ED [x ] I -I , and F= F El) [y] .I ,. ·Hence 1 a. O'..C - 1 ex. m:: - 1 for x c E, x = xl © r A.X .. and 1 icI-I' i·

(x) = (fa.(x + A..'x. ) ) I ~. 1 L 1 1 m.: icI-I'

By similarity of (X ) TI and there exists a c a. ac..1.. (yo.\x.cI'' Y1 Fl I' such thn.t fcx.Cx ) ga.(y ) for a. c l.~' • Clearly for a c I-I' f (x) 1 = 1 ex. - fcx.(x + L A. .x.) = f (x ) + ;\. _. Define y = y + L A..yq 1 1 1 0 1 1 1 1 i~r:..r 1 o: · icI-I' r- 4' Observe thnt g ( A..y.) ... ( ,/__.,'\ A.X.) for all <2' c I. a. 2- 1 1 = a. 1 1 icI-I' icI-I' Hence p (x) = (fa.(x + ;.\.x,)) 1 I.. 1 1 a.cl icI-I 1

(fa(xl) + f(LA..x.)) I = a. 1 1 a.c

= (g (y, ) + g c y.)) a - C/.. I>....1 1 U.CJ.

1 ,t (E ) S '\V(F ) • = ''+ (y). Hence :.r u v

By a sirnil.<:r argument one gets "'-\/(F f ) • Thus ,.t (E ) == "f" (F ) . v ~Chf ~Eu 'f u v i.e. (x ) ... is similar to (y ) r· By T'm::orern 2 of this chapter ex. ac~ a. ac there exists an isomorphism T: Ji; --t Ji' such thnt Tx = y for· u v Cl. a. a. c I. T clearly extends T si:'.lce tr..ey coincide en a total subset of E• 1 1 31

As another application of 'l'hecrem 2, vie obtain the follmJing characteri?.ation concerning similar bases.

Theorem 1+: Let Eu' Fv be B-·complcte barrelled ·spaces with u, v defined by the families of seminorms (pi\cK and (qA.\cL respectivly.

Let (x ) . I' (y ) I be total g8neralized bases in E and F a. a.c a. a.c u v respectively. Then Cxa.)a.cI is similar to (ya.)a.cI if and only if for any i c K there exists A. c L and M > 0 such that m m (a) .a x an. y. ) <[ n i ( I:- 1 n==l n n=l n and m (b) - a x. )

Proof: For the necessity, assum~ (x ) is similar to a. o:.cI

By Theorem 2 there exists an isomorphism T: E -~ F such that u v Tx == y for a c I. Hen·~e by Propo.si.tion 4, Charter 1, (a) and (b) (J, ex. are satisfied. For the s11fficiency, a.:_rnume (a) and (b) hold. I.et G be the linear span of (x ) with the i·el

T is linear. By (b) and Proposition L; of Chapter 1, T is continuous.

Hence T can be extended to a. linear map T: G -) F (see (29) v Chapter 3, § 8)) which iG cont:i.nuous. T is one-one on . G since for

n n x c G, x /.... x. and ir:rnlies "\ A.. v. = o. = L J. 1 ·' . [_ 1"1 i=l 1 32

n Hence g ( [ A.. y.) == 0 for a c I gives /... = 0 for 1 .:S i .:S n. a.l ii l.

(f ) · be the functionals associated with Thus x = G. Let o:. a. .c I ·c go:. )a.c I m respectively. Then for x c G, x a X. and = £ n i n=l n m g.(T)=g. ( _[ a Yi ) = a. = f. (x) for all i c I. For xcG= E J. x 1 J. J. u' n=l n n x =lim z~, zp c G. Since fa., ga. and T are uniformly continuous, g (Tx) = g (T lim zr) = lim g,. (Tz,) - lim f (z,,) = f (x). Hence a. .:.- a. 0:. a. p ~ [3 ~ ~ [3

f (Eu) S 'o/(Fv)• By repeating the argument and using (a), we get "f"(Fv) S p(E) • Therefore p(E) = ~ (Fv) and the bases are similar.

Remark: It is worth noting that Tt1eorem 2 of this chapter cannot be strengthened to the case of complete barrell,,~d spaces. As an example consider the following: I,et E be an infinite dimensional Banach u space with (x f ) a generalized basis and let F be E with u' a. o:.cI v the strongest locally convex topology (13) (Chapter 2, g 4, Example 3) •

Then E and F are complete barrelled spaces py (14) (Chapter ·4, S 1 u v 1

Proposition 1) (x. 7 f ) . I is also a gen~ralized basis in }' and a. a. a.c v is similar trivially. Hm·iever E and F' are not isomorphic since u v v is strictly finer than u. (ll~) (Chapter 1+, £ 1, PropoEdtion 1).

Th~orem 5: Let be a B -complete 'l'VS and F barrelled. Let r· v f: E ___, T!' be a. linear onto map with closed graph. Then f is open. u v

See (11+) (Chapter 4, £ 4, Theorem 7). 33

Definitfo:'ls: Let J be the class of all barrelled spaces. A

locally convex TVS E is a B( J ) [B ( 'J )J space if for each locally u r convex space F c a linear continuous and almost open map v J , f: E --} F onto is open [ a linear continuous one-one almost open u v map f: E -7 F onto is open ]. u v

Theorem 6: (T. Husain) Every barrelled spnce E which is a B ( u r J) space is B -complete. r

Proof: A proof may be found in (14) (Chapter 7, S4, Theorem 5).

Definitions: (Mcintosh) A locally convex TVS E is a (C)-space u [resp. (C )-•space] if every linear subspace [resp. dense linear subspace] r D of E' (c>= a-(E', E)) whose intersec-Gion with each c-(E', E) bounded subset B S E' is cs- (E', E) closed in B, is necessarily closed in E 1 tS" •

We note the

(resp. B -complete) space is a (C) (resp. (C ) ) snace. r . r

Theorem (Mcintosh) Every barrelled (C)-space (resp. Cc )-space) z: r is B~complete (~esp. B -complete). r

Proof: See. (22) (Page 399) •.

Remarks: (a) Closed graph theorems have been proven for B ( 1) and r (C ) spaces, (14) (Chapter 7, $4, Theorem 5) and r (22) ( '11-,eorem 3) • However an attempt tc extend Theorem 2

of this Chapter to BI' ( or (C ) barrelled spaces :J ) r does not lead to a generalization as Theorems 6 eJld 7

show, so in this sem:e Theo:cma 2 i8 most general. (b) Theorems 2 and 3 of this Chapter also hold for barrelled

Br(J) spaces (in particular for barrelJ.ed Br-complete

spaces) because of the closed graph theorem for these

spaces and since a B -complete space is c6mplete. r

}. Inductive limits

~beorpm 8: (Robertson and Robertson) Let E be a generalized strict u inductive limit of B-complete locally convex spaces and- F an v inductive limit of Baire locally convex_ spaces. Then.

(a) Any linear continuous map f': E -7 F onto is open. u v (b) Any linear map g: F --7 E with closed gra;;ih is v u continuous.

See (26).

Theorem 9: Let E and ...Ti' be generalized strict inductive limits u v of. B-complete Baire locally convex spaco.s. Let (x ) be a generalized a. a.c 1 basis in E • If T is an isomorplcism T: E ---} F suc::i that u u v Tx: --y,.,,cx.cI, then (y ) is a generalized basis in F and is .a. v. a. o:.cI v similar to (x ) Con-,rer:>e ly if (y ) is a g0 eneralized_ basis in a. cx.c r· a. a.c I m. F similar to (x ) I' then there exists an isomorphism J.. E -~ F v CX. CT.S u v such thnt r;:'x = y for all a. £ I. ex. 0:.

The proof is essentially the same as that· of 'l'heorem 2 section l of this chapter except_t.tat one must apply Theorem 8 a~ove, instead of

Theorem 1. Q.E.D.

Let be the generalized. strict inducti-.re limit of a sequence (E ) ~ of locally convex spaces, nnd let (x ) be a n nclf a a.cI family of po ints in . E • 'I'hen 1 35

(a) If (x ) I is a total gencralizt:d -basi.s in each E a. w: - n' then (x ) I is a total generalized basis in E • a. a.£~ u

(b) If (x ) I is an extended Schauder basis in each E a ac n'

then it is an extended Scha<;der basis ir1 J:'J • Extended -· u basis meaDs an arbitrar;:r indexint; set, the basis expansions

to be carried out according to some fixed linear ordering

of the· indexing set.

----Proof: See (1), (Theorem ?).

Theorem 11: Let E F be strict inductiue limits of B-complete Baire ------u' v locally convex spaces, the topologies defined by (p.). K and l. l.£ ' Let (x ) arid ( v ) be total generali£>ed bases in E F a. O'.C I . v Ct Ci.£ I u , v

respec tivcl;'/. T'rien (x ) is simila1· to (y -) ~ if and only if for a. cu:. I 0:. cu: J.. any i. c K the::.·c exists a A. c L, H > 0 s:ich that m m (a) ( a x. ) < M q. ( a Y. ) pi [ n J. A 2__ n - 1 n n n=l n=i

m m ..., and (b) ( I- .C:. ) < H ( !'!. x. ) qi n Yi 1:';\ [ n 1 .c. n n:::l n=l

hold for all finite sequences ..• , i of indices in I and scalars m ..... "' m.

Proof: Assume (x ) I is similar to (y ) r· Then by Theorem 9 a ac a uc this chanter· thG:!:'e exists a1: isomorphism ·r: E --··~ l" such that u v

Txa. :::. yCl, a. c I 1 and he21ce we get (.oiJ and (b). The converse follows as in 11\:leorem 4 of this chapter. Remark: Because of .Theorem 10, one can get theorems analogous to

Theorems 9 and.11 by letting E F to be generalized strict inductive u' v limits of (E ) N and (F ) N where Cx ) and n nc1 n nc ex. ex.cl total generalized bases in E and F respectively. The proof n n would be the same, except for applying Theorem 10•

•L~. Dual generalized bases and further ~ffolica tions

Definition: A dual generalized basis in a TVS E is a biorthogonal u

The next th.eorem due to Klee answers the question of existence.

1'heorem 12: (Klee) If E is a separable locally convex space, then u there exists a dual generalized basis for Eu·

Ir.2.Qf: Since E is separable, there is a sequence (y.). N such u l. l.£ that [y.] == E • Without loss of generality, let (y ) be such i u ~i icN that every finite subset is linearly independent. Let E be the linenr n span of Let 'i'hen there exists tY1' .•.' ynJ • continuous linee.r functionals f , f such that f (x ) = r Cx) == 1 1 2 1 1 2 and r Cx ) r Cx ) o. Also, we observe that r , f have continuous 1 2= 2 1 = 1 2 linear extensions to all of E ' and f. (x.) = oij for i, j = 1, 2. ,.. u J.. J -- - "I Recursively, let t_x , be such that 1 ... ' x n ' fl' ••• ' f n..,,> ( "\ E' sp LX1' • •. , xn J = n and fi (xj) = J ij for 1 _;S i, j ~ n. The biorthoaonality gives txi' ... , xn"J linearly independent. Now define n x.·. Linear independence Cif {x , , x I 1 1 ... ~{ n ' n+lJ i=l follows from the biorthog?nality, and .~., x, x J n n+lj

As abova, there exists such that fn+l(x) == 0 for X E E n 37

and f Cx ) =l. Also, f.(x.) = 6ij for 1 S i, j ,:S n + 1. n+1 n+1 1 J Hence by induction we can obtain a biorthogonal system (x., ~r 1 r:). 1 1!:1 di.lei I . for E such that [x.J = E , and we therefore have avgener<.1.lized u 1 u basis.

# Example: ·Let E be the Frechet space of all functions ai1alytic on u the open disc Iz I< 1, topologized by the metric of uniform convergence 2 n on compact sets. Define (x , f ) N. by .x (z) 1 + z + z + ••• + z n n nc n = (n)(O) (n+l)(O) and f by f (g) =Fi___ - ~.___,.:..-;... for n = O, 1, 2, ••• n n n: · (n+l)!

As we showed in Chapter 1, section 6, (x , f ) ~ is not a generalized n n nc1~

basJ_·.,,0 but si·nce · (x ) i·s tota1, i·t i·s a duaJ. generalized basis. c ' • n ncN

, . . . ---Remark: Dieudonne (see [16]), e;ives the fol101,11ng criterior. for a biorthogonal system in a TVS E to be maximal: (x , f ) is u a. a. a.c 1 1 ~ ]l maximal if and only if [x ] S: [f ] where LX c E': a. a a. = [r or· if and only if f) Ker f £: [x· ] • a. a. f a.

(Davis) ( ~ ) be A biorthogona1 system for a TVS \Xa. ' I a. ca: I E ~· Let tp be the quotient map ((1 E -4 E/Ker:.t ~ the coefficient u u 'f map associated with (x ' f ) 'fhen• a. o:. a.c r·

(a) ( Cf <~~ ), f ) is a genere.lized basis for .E/Ker ...t a.,., a. a.c 1 ,..., ':( where f is defined by f ( Cf (x)) = f (x). Cl. Cl. a.

(b) If (xa.' fa.)a.cI is a dual generali~;;ed basis, then

({f (xet.), fa.)cu:: I is a Mnrku.schevich basis for J:;/Kerii>

with the quotient topology. (c) If (xa., fa.)a.cl is a mnximal biorthogonal system for

Eu and Eu is locally convex, then ( q> (xa.), fa.) m::I is a Harkuschevich basis for E/Ker p if and only if

(x , f ) I is a dual generalized basis for ex. ; a. cx.c Eu •

Proof: (a) Let Ji- denote the range of ~ Define ~- : E/Ker 1 -} J.1 This is well defined since lp is onto. Le t ~ have

the topology induced from E/Ker 4' by ~. Hence 4' is continuous with respect to this topology. Also, f is the coefficient mop associated ,J with the continuous linear functionals ( f' ) n and ex. a.c .L Hence ~ is one-one and ( {() b: ) , f ) I is a generalized basis. '"f a. a. Cl.C (b) Since (xa.)a.cI is total and {f ~ linear continuo~s

onto map. ( IT> (x' ) ) I is total. Combined with (a) this- implies (b). ' t..f a a.c

(c) Assume ( (() (x ) , f ) T is a .Markuschevich basis. , a. O'. CX.C.L Hence ( lp Cx))a.cl is total. Since (xa.\.r.cl i.s maximal, by the 1 previous remark, Ker S[xa.] , so for f c f c (Kerp )1. Hence i [xc. ] ' ,_;;. ,,..., by. each such f induces a map f: E/Kert '"--1 ~ f - f "tf There fore f(x ) = 0 and since C{f(x)) I is total we h:..ve a. a. Cl.£ ....,· f = O. Hence f = 0 and so (x ) is total and therefore (x , f ) T a. a.c 1 a. a. ac- is a dual generalized basis. Q.F~. D.

'l'he_orem 13: Let E F be B-complete barrelled spcices and u' v (x f ) (y g ) similar biorthoswnal systems. for E , F a. ' a. cx.e:: I' - a. ' a. cx.c I ~ · u v respectively. Then there exists an isomorphism T: E/Ker,f --} F/KerY such that T( lp (xa.)) = Cf '(ya.) for a. c I, VJhere {f , 4'' are the quotient rnaps and

Proof: By part (a) of Lemma 2 this section, ( IG (x ) , f ) I and '1' a. Ct m:

respectively. These ,bases are similar since (x , f ) I is similar a. a. a.e:: -· -.., to (ya.' ga.)a.e:I' because for p,Y the coefficient maps associated ..._, ~ with the above and ~ s E/Ker~ , we have ~ (~) =- p (x + Kerf ) = ,.., (fa. (x + Kerp ) ) a.cl (f ( lp· (x))) I= (f (x)) ~· Similarly = a. a.s a. a.e:: J. Y(j) :: (g (y)) r· But by similarity, (f (x)) I:: (g (y)) r· a. a.e:: a a.c a a.e:: Hence f (~) = "'? (y). and therefore f (E/Kerp ) 5" Y (F/Ker"i_,. ) • ...... '"""...... ,....; Similarly. ~(F/Ker't') !;;. ~ (E/Ker ~- ) , and we obtain "t'''

~ and l\.f" are continuous~ so Ker d>.... are closed subspaces of E , F respectively. 'I'he quotient with a closed subspace is hereditary, u v so, E/Ker ~- and F/KerY are B-complete barreHed spaces \·tith similar generalized bases. Hence by Theorem 2 section 1 of this cha.pter, there exists an isomorphism •r: g/Ker ~ ---7 F/Ker"'.Y such that

T( Cf (x )) = /r;t(y) for all a. e:: I. Q.E.D. a. I a.

Remark: Since the cJos~d graph theorem 2~..t.11da for B -complete barrelled ---· r spaces, the above holds for same.

Lemma 3: Let (x , f ) _ be a biorthogonal system for a 'rVs ...,p and a. 0:. a.e:: J. .u 1 13 J: E --7 E 1 P . the canonical map. 'l'hen if (x f ) is a dual u ex.' a. a£: I generaliz.ed basis for E it follows th::1 t (f , J(x )) I is a u' a. a. a.e:: generalized basis for E .~

Let (x , f ) I be a dual generalizEi"d basis for E For a. a. a.t:: . 40

f c n Ker J(x J(x )(r) -· f(x ) = 0 for Ct c r, and since O'. ) ' a. a. a.cI

[x ] E f Therefor~, since J(x )(fs»= ::. a. = u' = o. a. . f f3 (xa.) 6pa.' J(x )) is a eeneralized basis in E '13 (fa. ' a. a.cI .

'l'he above lemrnR justifies the term "dual".

De finition: ... " biorthogonal systems for E , F respectively. are the u v If ~ ., ·"f' coefficient maps as::wciated with (J(x )) -1' (J(y )) I' then the a. a.c a. a.c "' " systems (xa.' fa.)a.cI and (ya.' ga.)a.cI are *-similar if . ~ (E') ='o/(Ji'').

Using the above definitlon, the foll.owing is a corollary of

'l'heorem 2 of this chapter.

Let F' be TVS 1s such thc;,t; "''13 F''f3 are B-complete "'u'"' v ..., ' barrelled. If f ) and ) are *-sir.1ilar dual (xa. ' ex m:I (ya: ' ga. cu: I generalized bases in E , F respectively-, then there exi0ts an isornar u v 13 phism S: E' ~ F' p, such that S(f ) = g , a c I. ex a.

Proof: Since (x , f ) I and ( y g ) ltre d'..1n1 r;en2rali.zed a. ex. uc: - c~' 'a. exsI bases, by Lemma 3, and (g , J(y )) I are generalized (fa.' J(xex))a.cI a. a. a.c

bases for E1 13 F'P !"espectively. Also, w~ observe that they are ' sir.iila:r· since (x , f ) and ry o' ) are *--similar. Hence by a. a we I ' ex' 0 0'. cu:: I ,. , . ,,. c E'rj -~""f~ Theorem 2. o ftnis cnc.pter, there exists an isornorprnsm .;:i: 7 .i:

such that .S(f ) ::: g , for all a. c I. Q.E.D. a. ex

The lemma holds for reflexive C.N.s., since they are :F'rechet~"

hence the strong duals are B-complete and barre11""'d~

We noH prove a theorem dealing with *-sfr&ilar bases which is analoGous to Theo~em 2 bf this chapter. 41

Theorem 14: Let E , F be reflexive spaces with (x , f ) I' u .v ex. a~

(y 0 • ) *-similar dual generalized bases for E , F- respectively, · ex.' 0 a. ca: I u v and such that E'~, F'~ are.B-complete spaces. Then there exists an

isomorphism T: E -t F such that Tx = y for all a. c I. u v a. ex.

Proof: Let JE' JF be the respective canonical maps of E and F u v with and F'~'~ Since E F a:::-e reflexive, . u' v isomorphisms. Since reflexive spaces are barrelled and the strong dual

of a reflexive space is reflexive, by Lemma l1- there exists an isomorphism

such that S(f ) g for a. c I. Recall that a. = ex. " .... where 1 , "t' are the coefficient maps associated with

(JE xc)a.cl and (JF ya.)CJ,cI respectively. Consider the conjugate map 1 (s- )*. Then for g c :B'', a c I,

[(S-1)* (JE xa) - JF Y) (g)

::: (S-_l)* (JE xa.) (g) - (j·r ya) (g) ... "" = (JE xa.) ( <£ -~ ~ ) (g) by -similarity of (x , f ) and (v Thus (S-l)*(JE xa.) : * ex. a. c:c I "a.' gcx.) a.sr

J"Ji, Ya: Hence cs-1)* I [JE xa.] = [JF ya.]• Now define T by .,. -1 -1 T = uF o (S )* o JE. Clearly m... i.s an isomorphism since each of the -1 maps JF'' S , JE are. Also

T -1 ( -1) J 'j ( ) Tx ( 0 s * a. = .., F ' " E' xcx.

=J -1 for ex. <:: I. F

Theorem.15.: Let E F be reflexive B-complet.e spaces such that u' v E'f3, F'f:l are B-complete. If (x , f ) is a generalized basis Cl. CL CX.C 1 systern which is simulta.neously s "cmilnr and·* -similar to a dual generalized 0 bas. i·s (y g ) J n F +he.n therA exi"sts an isomorphism .. cx. ' .ex. cx.c I .• v ' v

T: E -7 F such the,t Tx for ex. c I, and both are Markuschevich u v a. - yCt. bases.

Proof: Since E , F are B-complete and barrelled, and (x , f ) I u v cx. a. cx.c is similar to (y , g ) . I' hence by Theorem 13 of this chapter, there a. ex. a.c exists an isomorphism. S: E/Ker ~ ~ F/Ker''f" such that S( C(7 (xcx.)) =

· ' ( y ) for ex. c I, where '-fl'o, are the respective quotient maps cp cx. {f' and ~,~the coeffiecient maps •. By Lemma 2 (b), ( Cf'(ycx.))a.cI is a 1 1 Markuschevich basis for F/Ker"t" Now s- ( tp•(ycx.)) = s- (s( lp(xcx.)))

Cf' (xcx.) for all. ex. t I. Hence ( {P(x )) I is a Markuschevich basis I CX. UC for E/Ker ~ But (xcx., ft"t) ex.cl a generalized basis gives Ker if hence E/Ker is isomorphic to This gives (x ) I a M'J.rkuschevich 1? ex. ac basis for E • This implies (x ) is a dual generalized basis. By u a. a.cl •-similarity of (x f ) and (y g ) and Theorem 11+, we get a' a. cu;:I 0:. 1 a. a.cl the desired isornorph~s~ T: :8 ---} F such that Tx == y for all u v c.. ex. ex. c r. Q.E.D.

Coro,1].ary:· Theorem 15 is true for the following pairs of locally convex spaces with the same hypothesis: , (a) '.vnen E, :F' are reflexive 1',recJ1et spaces

(b) When E, F are reflexive Banach spaces

I' (c) When E F are Moritel, Frechet spaces '

'ine proof is trivial since these spaces aru reflexive, B~cor.i-:::lr;te and their strong du:-:J.1.s are B-com~)lete. Q..E. D.

Rergark: In regard to *-similar base?, :it has been obse:cved by o.rr. Jones that simil-'ir Sch::ludcr banes in ba:-ael1ecl spa.-; ea are *-·sir.Li.lnr. CHAPTER III

Countabl.L ba:crelled spaces and bases

1. Countably barrelled spaces

Definitions: (a) Let E be a locally convex TVS. E is countably u u barrelled if each ~(E', E) bounded subset cf E'

which is the countable union of.equicontinuous subsets

of E' is 'itself equicontinuous (Recall that for Eu ' c Fv two 'l'VS 's, H - 'd...~ (Eu, F v ) is equicontinuous if for

ea.ch neit;hborhood V of g in F , is a v n fdI

neighborhood of ~ in E ) • u

(b) Let A. c E be a subset of a ·r11s. The polar of· A, - ll denoted A0 is fr c E': lrCx) I < 1 for all x c A} • '-·

•oo TIIB bipolar of A, denotf:d h is { x c E: If(x) I .S 1

0 for a.11 f- c A } •

----Remark: Countably b~rrell2d snaces were introduced and studied by T. Husa:i_n [15]. Mo~;t of the results in sccti:ms 1 and 2 are due to

'J1. Huc,;11.:in and are given here for the sake of completeness.

Let A, B, ( A' ') ~ be subsets of a locally convex CJ. cu: .L TVS Eu· BC Ao (a) If A £ B, then :2 •

J A.-lAo. (h) If /\. 1· o, (A..Ci.)o -· ' i (c) ( A )o -· .A:.i v a. n a. a.E::I a.e: I l+l+

(d) For each set A, AO is a ts"(E', E) closed convex .set

containing Q.

(e) If A is circieci, so is AO .

Proof: See .(llf), (Chapter 2, ~. 8, Propositicn 12).

Theorem 1: A locally convex 'l'VS E is countably barrelled if and u only. if each barrel B which is the cot<.ntable intersection of convex, circled closed niee;hoorhoods of the origin Q is itself a neighbo:chood of -Q.

Proof: :for a proof see (15), ( ~ 3, Theorem 1).

The followin.~; provides numerous examples of countably barrelled spaces.

Corollary 1: Every barrelled 'I'VS is cuuntably barrelled. In particular,

/ every Frechet space and each Banach S:;_)~'tce is countably barrelled.

See (15) ($37 Theorem 1, Corollaries 1 and 2).

Theorem 2: (T. Husain [15]) Let E be a countably bari:·elled TVS. u Let (f ) be a o-{E', ?.~) bounded sequence of functionals in .E'. n ncN If (f ) "T converges to a linear functional f pointw:i.se, then f c E' n nc:1· arcd (f ) 'r converL_"es to f uniformly' on each precompn.ct subset of Eu. - n n:;:;h

Observe that each singl•:?ton L( f n"~is· eouicontinuous,... - · and . ...._ ncN t :'..s bounded by assumption. Since J

E is countably barrellert, is eq_uicontinuous, and by (2) u u n=l

Fropo:sit ion 4) ' f 1: E'. by (2) (Chnpter 3, $. 3 Proposition 5), (f) •r converges to f uniformly on each precompact n nn~ subset of E • Q.E.D. u

Proposition 2: Let E be a countably barrelled TVS, ·and F a u v locally convex TVS. be a sequence of equicontinuous sets (]) of linear mn.ps of u into F such that H is simply bounded. "'u v U n n=l 00 Then u H is equicnntinuous. n::.:l n

·Proof:· . For all -:i1 t N~ ''f CH is -linear arid corftinuous, h-enc e the-re --- n n exists a unique transpose f': 1~ 1 --> E', defined by f (x), y' > =· n < n

x, f' (y') > for x c y' [; F' f (x), y' > simply means < n Eu ' • < n

y' (f (x)) • Let H' = ( I-r.• f c H 1 Since H is equicontinuous -n n n n n J . n

an·a y' are continuous linc:ar maps, \·!e have for each equicontinuous f'n'

1 I subset H' c. F' . H' (H') - f f 0 y': f' c T1' y' c M'J is oq uicontini10us ' n- \. n n ""n' CJ) ro in_ E' Since H is simply bounded, H 1 (M') is E) . n n en'E' . ' n=lu n=lu

bounded. '!.'herE:fcre s:;nc8 E is countably barrel}E'd~ H' (H') J.S u n

Q) an equicontirnlo;is subset of so H is equicontinuous. E'' u n n:..:1

Corollary__..,_1: I.et E be a countably barrelled and F · a locally ___ u v -convex 11VS. If (f· ) N - is a simply b6urL1ed sequence- of cotit2.nu01~s n nc lj_near maps of (f ) is equic,,nt:inuou.s. into then n ncN

Proof: :follows i'rorn Preposition 2 by taking H for each n. --- n

Let E be a countably barrell0d and F a locally -U v

convex TVS. If (f ) m:N is a sequence of crmt:inuolJs linf~ar mnps of 11

E into cnnverges pcintwisc to a map f: E --~-11~ u u v'

th2n f is coRtin~ous and linear. 46

Proof: Trivially f is line3r and ( f ) •., is pointwise bounded. n nch ;By corollary 1, (f ) N is cquicontinuous. H~nce by (2) (Chapter 3, n nc S3, Proposition 4) f is continuous. Q.E.D.

Proposition 2= Let 'E be a metrizable locally convex TVS Then u 4 E_' 13 _is countably barrelled.

Proof: See (15), ( £ 4, Proposition 1 and 4).

2. Permanence properties

'I'heorem 3: I.et (E ) _ be a family of co; ntably barrelled spaces ex. m::.1.

and (fcx.\(.cI a family of li::icar maps into a linear space E. Let E

be endowed with the finest locally ccmve~{ topology u such thctt each

f is continuous for each a. c I. Thim E is countably-barrelled. ex. u

See (15), ( S8, T}1eorem 8).

Let CF~ ) be a :family of coi;ntilbly barrelled .spaces ex. O'.C I and E it's induc1:ive limit. Then E is also counb.bly ba:::-relled. u u

-·-Proof: This is just a special case of Theorem 3.

be a COl'.!ltably barrelled ·-:>pace and M S E a

closed subsp::i.ce. Then the quotient E/M is also countably barr-eJ.led.

See (15), ( S 8, Theorem 8, Corollary ll+).

Corollary .2: Let (Ecc.)cx.cI be a family of countably barrelled spaces.

'I'hen its diiect sum is also counta.hly bn.:tTeJ:led.

---Proof: See (15), ( "~ 8, rl'heorem 8, Corollary 13),, 47

In chapter.II, Theorem.2, we showed thnt the isomorphicm theorem

did not hold for genernlizecl bases even if both spaces were complete

and barrelled, however it didhold for B-cornplete barrelled spaces. In

the case of Schaudsr bases the theorem can be much improved, as the

·follm-1ine; shm.rs.

Let E , F be countably barrelled spaces and let u v

(xn ) nc N~ is similar to (y ) if a,nd only if there exists an isomorphisr:1 (xn ) nc1···~i n ndT T: E ----'r F' such tho.t Tx "" y for all r, c N. u v n n

rr. i <~ Proof: If •> v.D isomorphis~ SU8h th;-:t ·rx - v for all n c N n ~n ' co CJ) ro ( .--. then clearly if a x converges, the!l T a x ) ~ a ,, L n n > . n n - '-'·-' ..-· n .J !1 n=l-- n=l n=l also converges and vice-versa, hence cme gets similarity. }'or thr;: conversP., assume the 'bases 8."e similar. Let (f ) N ~. E' be such that n nc ro

is a biorthogonal system. Then for x= .!."" (v).... ~ x.. • x £ l:u ' z n n n=l

Define T· by m -m ..---. 'r (x) - ' f (x) m -- ·1, 2, m L- n yn ... ' n=l and T by Q)

T(x) :::; f (x) y • [ n n n=l '.!' is 1trnll def: n'_:d, since by similarity of the bases L .. is .. n==l The T 's are cm;,tinuous and linenr .sin:::r:2) f ndJ are m n ...... --cu 'I'x ::: G f (x) v ::: and since continuous nnd. 1inetu·. ir:.plies n vn a, n=l'- 48

f (x) = 0 for all n c N, hence x = g (yn)ncN is a Schauder basis, n

because (f ) N is separating. Therefore T is one:--one. 'l' J..L'> onto n nc,

CD CD - since for -y c F , and by similarity b x v y= ~ L n n n=l

Q) f (x) x converges in Fu for some x c E. Also 'l' (x) ~ T(x) L_ n n m n=l

'Ii, for x c ..:J by definition. Hence (T ) is pointwise bounded on m mcN E and T :i.s in the poj_ntwise closure of (T ) By Propositic:n 2, u Ill r:icN• Corollary 2 of section l of this chapte:c, T is linear and continuous.

Likewise T-l is also continuous. Hence- T :i.s thedesire isomorphism

E onto F . u v

Remark: In view of the facts that ir1duc tive lir.1its, direct sums and

products of co;mtRbly barr(~llecl. srmces are ulso countab~ y barrelled,

ths th-,)Orern nold3 true .for thet>e C<:1SOS,

Let },,._, be a countably barrelled ?.VS. Then every weak u

Schauder basis in E is a Schauder bor-;fr; fo1~ ~he in:i_ tfal · to:Pology u. u

Proof: 1-e t Cx , f 1 be a weak Schauder basis. Hence --·-·- n n'ncN ( x n ' f n ) nsH~ is a W•3ak Harkusche:vich basi.s, so [x J_ = E . in the initial topolor;y n u

by ( 1) Theorem 10. For P.ach x c· E, choose a se::r.;.ense (y ) ~ E n ncN

such th.qt " ---} x 8.r? c1 y c .s p f x . : l < i < n ~' • Consirler the -"n . ' n - ~ 1. . - .,..,. -J n SEOquence (T ) of T : E -~ E by 'l' Cx).'"'"' f. (x) x. n ncN n n L ] 1 i=J. for x s E, n = 1, 2, Ohserve thP.1.t lim 'I' (x) - x ir: the ... n ( ,, n -->ro HP.-.nce ; rr (··)' v:eak ·topology .. -· • ·, -· .A f ~r is ueakly oou!'!ded, hence bounde:i ir: t r.. .i :r:.c.· (' the initial topology. By Pro~osi tion 2. .9 l of this ch0.pter, ('I' ) M n nt.n is equicontinuons in f;hc ir.ita.1 tc•polc1gy. Hence x x + lim 'l' (x ) - i;. y-n n [_ -· x + lim f. (x -. y n)x. n i

:::: li.m v ,.i..12TI ·. ( f. (x) x. ) "n f~ 1 J. - yn n n i,Sn

= lim [_ f.(x)x., J. 1 n i~n and so (x f ) is a. Schaucier basis for the initial topol.ogy~ n' n m:N - · Q.E.p.

De fi::-1 i.tion: Let Cx , f 1 be a basis in a locally convex TVS E ----- n n·rn::N u' u defined b;y 'I'hen it· is monotone if for each ( {1 .... fixed but arbitrary, p\ / f. (x) i~ a nondecreasing A. L :-1 'l xi? rn::N 'i= .I

l1en:E·1.:.~ 1: I.et. E be a locally con'rex TVS and f a linear functional ----- ·u li' on ~,. Let g: E ·-:> E be given by g(x) :::: f(x)a, a £ E, a arbitrary, and a I- g_ •r1wn g ccntinuous implies f is con-'.;inuou<;.

Proof: If f is not continuous, then since f is 1.inear, f is not ccntini-:ous :.:1t the origin 9. rience there exists a net (x , I 0:. 'w::. with :;\x. ~ g co,r1q f.Cx) +o. Assrm10 jJCxo/ I 2: c..> .o. Now g ccr:t:inuous gives e;Cx. ) ---} g(G) :: o. Hence a = e'..:(~) '-/ o. 'I'his a f(x ) 0 gives a = G. Contradiction, hence f is continuous. Q.E.D...... ProDcsi ti.on 1-t: .l.J.. E is a locally convex TVS, u given by ---i..--.--...- u

1.' and ~J hci.s a mcnotone basis, (x , f ) .,, then the coordinate func tionnls u n n nc1. 1 are cc1:.tim1ous and h,2nce the basis is Schnuder.

_,.,. I:'.':c·oof:___ _ Let n s N arbitrary. Then define u•"". by 50

g(x) = f (x)x • For any A. c L, n n n n-1 Pt.. (g(x) )__ =.I\ ( f. (x)x. - ,. f.(x)x.) r= 1 1 .t,__l 1 1· :i.=l i= n-1 f. (x)x.) + p, ( r.;- f.(x)x.) < 2 p,(x). 1 1 II. 1 1 II. "i=l

Hence g is continuous, so f is continuous for each n by the n above lemma. Q.E.D.

Theorem 6: Let E be a locally convex 'l'VS, u given by (p, ) /\.cL. u II. Let Cx , f ) N be a basis in E. Ti1en if for anJ; A., n there n n nc exists M > 0 such that Pt.. (s (x)) :S M pA. (x) · for all x c E, 11 n sn(x) = L f. (x) x., then is a Scha.uder basis. i.e. i=l 1 L

f · is continuous for each n. n

Proof: To do this we define a new family of semi norms p! by /I. n TI1is is well defined sin:e (x , f ) p~ (x) - supp, ( L f.(xh.). n n ncN n I\. ·1::::..• 1 J. 1 . having a bnsis it follO\·JS s (x) ~ x &nd p, continuous implies n . II. Pt.. (s (x)}--} p.\(x) for- es.ch A. c L. Hence ::mp I\(s (x)) exists. 11 11 n We observe the follo•·1ing:

··. - -· P{(Q) ~sup PA (s (g)) = O, . n 11 n f p~(x + y) $Up I, f.(x + y)x.) -- PA. ----L 1 J !l . i=:l n n ( [_ < sup PA fi (x)xi) -i- sup_ P/.. ( .r· fi (y) xi) i1 i=l n 1=1 ·

= p~(x) + p~_(y), and fo:· a. a scalar,

McMASTER UNIVERSITY LIBRARY 51

n p~(ux) = sup p~ ( .L f. {a.x)x.) . n I\. 1 1 i=l

Observe that (p~\dJ. and (p.\ \cL are equivalent families of semi norms since pA (sn(x)) ~ pA (x) itiplies that for any c >. O, there exists N(c) such that for n > N(c),

Taking s_up we gel

-c + pA(x) < sup r~?r Cc.)

Since c > 0 was arbitrary, let <: -·-7 O, hence PA (x) ~ PA (x).

Also, by assumption, for any A., n there exists M > 0 such that

Pt... (sn(x)) :S M Pi, (x). Hence sup pA. (sn(x)) :5 H pA. (x). Therefore n p~ (x) ,:5 M PA (x), and :'le:'.'1.ce the two families are equivalecit. Also,

(x f ) is monotone with respect to I I) since for A arbitr<.;.ry, n' n nd-{ \PA. A.cL' m-1 n. n '("" . p' ( [ ((.. x.) ·- sup C[' rx. x·.) :::. sup ( ex. . ;{.) ), J. J. PA 1 1 pi\. L J. 1 i=l ·n i=l 1_91~-J'!J-l i-1-~

n m < sup PA ( L et..x. ). = PJ (.L a..x.). l,5"'1.$m i=l l. 1 . .. i=l J. J.

By applying the previous proposition, f 's are continuous for all n n. c N. Q. E. D.

If E: is a normed space theii the theorem holds.

In his paper ('() Gelbaum examined the following problem. Let

(x . f ) be a Schauder b3.S is in R B;in2.ch ·space !.~, and (r, ) ~- n · n· ni:N n

any sequence of scalars. Consider a. x. for each n. J. 1

When does (yn) m:N become a Schauder basis? Here we present a

generalization of his result to the case of locally convex sequentially

complete spaces (Theorem 6).

Definition: A TVS E is sequentially complete if any cauchy sequence u in E converges. u In metric spaces sequential completeness and completeness

coincide, but not in general.

~ Definition: Let L x. be a series in a locally convex TVS ·E , u i::.::l 1 u 00 given by Then L x. is absolutely conver;;ent if for any i::.::l 1

A. c L, p,(x.) <+oo. -1\. 1

~: In general,· absolute conver1~ence and c nvergence are not

rel'tted. f'.. s e::amT,Jles consider first E = Reals 111i th tl1e usual topology. u Q) L is convergent but not absolutely convergent. Next consider n==l

(the space_ of al1 r~al sequ<)nces of ffoite support) with the nt'3. norrn of c Call zero scquencel3). Let 6 1, 0 n =Co, o, o, o, •.. ). ID .. L on . . 2 cmwerges absolutely but does not converre in E. The n=-1 n connection between the two notions is as follows.

Lemma 2: Let E be a sequentially complete locally convex TVS,. u defined by (pA.\_~L· T:wn absolute convergence implies convergence.

Q) Proof: Let 1-_ x. be an absolutely converr;ent series in 1 i=-1 53

m n TJ:~en for m > n, A. fixed but arbitrary,. pA. ( L x. ·- [ x. ) :: - i=l J. i=l 1 m m CD PA_ ( L x.) < [ PA_ (xi). . By assumption L x. converges i=n+l J. - i;:;;::;n+l i=l 1

m absolutely, hence \ p (x ) < c for all m, n sufficiently lar.:re. /..- -A. 1· 0 n+l · Since the space is sequentially complete, the series converges. Q.E.D.

Proposition 5: I.et E be a locally convex '11VS and (x , f ) 1\r u n n m:i. any countable biorthogonal system. Let (an)ncN be any sequence of n scalars. Let yn = [ a. x. for any n. 'l'hen [x ] = [y ] if i=l 1 1 n n and only if a 0 for all n. n f.

Proof: Assume a / 0 for all n <: N. 'I'hen and by n substitution each xn is "lXp:::"e.:-;sible in terms of· y ,. ••• , Yn· Hence 1 By the definition of y ' [y] ~ [x ]. Hence . [x] [y J. n n n n = n For the converse assume [x] = [y J. If a. = 0 for s0me k, then n n k n fk(x. ) = 1 anc fk(J' ·) = fk ( ,. a. x.) ~ a = O. for alJ n. This !c · n f;i 1 J, k

·implies xk i [x ] , for if [y J them· n n '1.:: = lim _ zn Z CS'J(X ) n t n and lim f, (z ) = O, contradiction:. Q.E.D. K n z csn(x ) n . ~ n ·

Proposition 7: Let Eu be a locally convex TVS, u defined by

and a subset. Th.'2n B is bounded if and only if pA. (B) is bounded for en.ch A c :.

--·Proof: See (30), (Crm.pter 12, S 1, 'i"neorem 2). Let E be a sequentially complete locally convex TVS, u

Let (x. , .. f.) . ; be a Schauder basis such tha.t J. 1 J.Cn11 for each fixed A. c L there exists M > 0 such that pA.(xi) i Mi\ for all i c N. Let (a.). N be any sequence of nonzero scalars. J. 1£ n Consider yn = [ a. x .• Then (y ) is a Schauder basis if and 1 n ncN i=l J.

Yn T only if P71. ( ) is a bounded sequence for each A. c .u. an+l ncN .( yn Proof: Assume (y )- . is a Schauder basis ( n ncN anc. i pi\ a --- ... n+l ) } ncN is not a bounded sequence for each A. c L. Then there exists A. c L

such that C ---}·CD for souc subsequence (n.) • We n. J. 1 Q) CD .~ x choose (n.) such that c > n. Hence < [ ni ) < J. n. 1 J. i.=l h 1 x l n. + ro. Thus, u:;_ th this choice of (c ), [ 2 < n. -2 n. 1 L c 1 ±=l n. J. is absolutely convergent. By Lemma 2, it is convergent to z, say. 1 Then f (z) - 0 if n In. and f (z) = - if n - n .• Hence n 1 ni ~en. J. 1 f (z) y n. n. f f ... 1 1 ) n n+.L = le --7 CD as n. -1· CD• Consider a +1 'I- n. 1 gn - a - -·--a • n. 1 n n+l 1

We claim that (y g ) is a biorthorconal family. Clea;~l v each g n ' · n nc N ~ J n is a continuous linear functional since each f is. Also n 55

r.t. n n+l g (y ) ( !Jl _fn+l J (y ) -- (f---- f ) a.x. n m = a a. l!1 z= 1 1 n n+~1 a n a n·+l · i=l

m .m 1 _l_ :::. a. f (x.) a. f (x.) --a L. 1 n 1 a 1 n+l 1 n n+l L i=l i=l

-- 0 if n I m and 1 if n = m.

Since a. /. 0 for all i, [x ] == [y ] by the ')reviou.s pro_position. 1 n n 1 Hence from Chapter I, section 7, 1heorem 11 (y , g ) form a Schauder n . n m::N basis. Observe that

n n fr•+l(z))C } (1) gm (z) ym = ,- f (z) x (:--··-- y L L m m an+l n m=l m=l

This follows by induction, since for n = 1, we have

fl f2 ) g (z) y = -· -- (z) (a . x. ) = 1 1 .( a a . 1 l 1 2

Assume (1) ho1ds for some n. Then

n+l n g (z) y - \ g (z) y + g +l(z) y .,. l .. m m L m m n n-. 1 m=l

.n -( fn+i (z) ' + _ fn+2 ::: [ f (z) x yn (~n+l J (z) ni m 0 Yn+l · an+l J 'n+l an+2 I m"°l

n+l f I n+2) = f (z) x (z) Yn+l• L m m -\an+2 . m=l

tJ)

Hence by induction, (1) holds for all n~ Observf: that (z) y L gm m m=l does not converge. If it did~ then since (x f ) · ·and (y g ) n' n ncN n' n ncN ·f · n+l ar.e Schauder bases; using (i), p ( --·- (z) y ) -i 0 as n -~ oo. A a n+l n

f \ In particular ani+l (z) Yni ) ---} 0 for the_ subsequence (n.) • . pi\. ( 1

n.i+1

Contradiction! lrence foi' each X c L, ·· l\· ( ::+l ) is bounded.

For the converse, assume pi\. ( : n ) is a bocmded _sequence of n+l n _for i\. fixed but _arbitrary. .Observe __ that since ___ (x _, f ) -- N is a n n nc basis, f (z) --4 0 as n --1 -oo, for each z c Eu. _ We claim tlnt n n \ g (2',) v --} z as n ---} oo. for z c E • :Jt,or ;\ c L, L. m "m • u m=l

g (z) y -· PA ( ~r:. f (z) x - z -( .:Il::..~\z) y ) rr. m ~ m m an+lf n. m=l

.n f (z) x (fn+l cZ,) < m m - z) + Pi'-. "Un I\( [ a +1 ) u!=l n

Since (x , f )_ N is a basis and is continuous, n n nc - z)--'.:\~ ' as n ~) oo, and since fn_(z) ---7 0 as h -; a:> and

f 1 (z) y J f Yn bounded-for all n, p ( _!1..:!:_____!}_ =-\f __.__ (z)\ F\ ! ;-- A. \ an+l n. J \ n+l n -) m. Hence Therefore (y g ) is a basis. I n' n m:N .m=l

Let E be a sequen~ially complete locally convex TVS, u u gi'ren by Let be a Sch-Huder basic; in E u 57

n such that (x ) is bounded. Then yn :xi is a basis if and n ncN = [ i=l. n only if PA ( ~~l xi) ncN is a bounded sequence.

Proof: Follows directly form the theorem by taking for all n. a n =1 n - Remark: 'I'he condition [ x. being bounded is a generalization 1 rn::N --- 1=. 1 of the notion of type P . basis due to Singer (28) •.

Proposition 3: Let f ) be a Schauder basis in a. locally .. (xn ' n ncN

convex· TVS v._. and (c any s~quencE; ·of vonzero scalars;; ···Then u n ) nc N (c x ) is also a Schauder basis. n n nt:N

Proof: The proof is trivial, since hv assumption, each x c E has u

([} b n a uniqu.e expansion b x := c ·x ·,.·and the uniqueness x -· E: n n 2- c n n n=l n=l ·n b \ of (b ) . gives the uniq.ueness of·the se_g.uencec ·Cn D~ n ncN · ( j ncN" Q.E. n

We can now present the following. corblla1'.'y to 'l'heorem 7.

Cs:rollary 2: If E is a .sequentially complete 10c-al1y convex· TVS, u :n :L x. i'=l J.. then ( y ) '1·'' where y = is a Schauder basis if and only n nc ~ ·n n n if where z = L x.' is a Schauder basis. n i.;::i J. ----Proof: Follows directly from Theorem 7 and the above proposition. Q.E.D. CHAP'I'ER IV

Normal cones and bases ia countably normed sp:=ices

l_ Cones

In this section we introduce the notion of cone::s and ,normal

cones, and give some fundamental facts about them. We then give some

results relating bases and cones.

Definition: Let E be a TVS. A cone· K ~ E is a convex subset u - u of E, wHh the origin G. a:s an extreme point i.e. A .convex set such

that

(cl). K+KS K.

(c2) o:.K s K, 0:. > o, a. re.e,l.

(c3) Kn (-K) - Q

· ·.Definition: An ordered linear space. E is one with a partial ordering

(i.e. reflexive transit.~ve and antisyr:imot:dc), denoted ~' such. that

Co,) x y implies x + z y + z, x, Y, z- c E _,_ < < u . . • 1 (02) x < y irnpliPS (Y.X < rx:r, for re2~. a. > o.

Re~_;ks: (a) We take t!'l.e llberty of denoting the usual ordering of the

reals by .::::;, but we ·,.;ill m-9.ke a distinction where confusion

may arise.

(b) Given a cone we can define a partial ordering

E < in u' by x .s y if and only if y - x c K, and given an ordered TVS E cine can define a ccne- K £ E u u by K - [x: 59

but since we are dealing with topoloc;ical vector spaces,

. _we give .the defini_tion in this C()ntext~

Definition:· (a) Given a locally convex TVS E with a separating . u biorthogona.l system (x ~ ) we can define a cone n' ·-'-n m::N' K(x , f ) = f x c E: f { x) .2: 0, n c N} . · That this is n n l n . . . a cone and is closed follows trivially from the.linearity

' and continuity of the This cone is called the (fn ) nc . N"

_cone ass9yi1;t ted. with the biorthogonal. system (xn , f n ) ncN• When there· is no possibilit:• of confusion we shall denote

K(x , 'f.) by · K for convenience. n n

(b) Given an ordered TVS E . with ordering < c E : u' -·' u is called the positive cone.

Observe that the cone K(x , f ) defined above is the positive .n n cone with respect to the ordering giYen.by x .$ y if and only if

f(x) < f (y). for.all n-e N. n - n ·

Defin:i.tiOn·: Let A c E be a subset of an ordered 1'VS E K the --- - u u' positive cone. The f>.lll hull of A is clefined to be FH(A) = r ' n tz c A: :x .S z .Sy, x, y s A j i.e. FH(A) :.:: (A+ K) I' (A - KL If

A -· .F'H(A), \.'e cnll A full.

Definition: Let .... be an ordered 'l'VS with K CE the positive cone. u -- u Then K. is. normal in E if there .exists a neighbm~hood basis g u of conF>istir:g of full sets.

If E is an or.dere.d TVS with KS E the positive u u cone, then the followine are equivalrint: 60

(a) K is riormal

.f>. (b) There exists a neighborhood basis of g consisting o~

sets v for Hnich g« x y,. y c v· implies x c < ' v. (c) F'o:r_ any two nets in E such that (xp) ~cI' ( y 13) 13c I u

convergence taken with respect to u.

(d) Given a neighborhood _v of _g in there exists a Eu ' neighborhood 'VJ of g in E such that g < x < y, u ·- y c W, implies x c V.

See (23), (Chapter 2, . ~ 1, Proposition 3).

· . ..,, Definitions: (a) · Given x, y c .6 an ordered TVS, y] c Eu' u [x~ = [z x,:5z.:5YJ.

(b) B ~ E is order bounded if there exists.· x, y c E u u~ such that B·f;_ [x; y}.

ProTJosition2: Let E be an ordered TVS, and K s E the positive · U U· · cone. If K is nor-mnl., then every order boun

Proof: See (23), (Chaph'r 2, b 3, Proposition 4).

Prouosition 3: Let B be an ordered localJ_y convex TVS, K c E u v u the positive cone. Then the following statements are equivalent:

(a) K · is normal in

(b) 'lhere exists a family (pa.)di.A of semi-norms generating

the topology u 51.l G.h tl1a t implies· (x) < n ( v) Pa. -• Cl. .J for all a. r. A.

--·-Proof: See (23)~ (Chapte"r 2, $ 3~ Proposition 5). 61

. Definitions: (a) I.,et E be an ordered _TVS~ 'l'h~n E is a vec;tor .. u u r y) lattice if fpr_ c;ny " y c SJIP. x, and inf [x, ~~-' Eu ' t Yi..., - J exist.

(b) A subset-· B c. }~ E a vector lattice i.s solid if . u, u

_fpr.. ~ <:; B and < for _s9me y c E gives. J IC I --- I YI . u' y E: B. Note, txL =sup _[x, -x1. (c) A linear subspace - M of a vector lattice E. is a lattice u ideal if N is a solid subset of E • u

Remarks: (a) Let E be an·ordered TVS· K cE the positive cone. u - ' - ·u If M E is a l"inear subspace. then M is an ordered ~ u TVS for the order determined by the cone K (l M.

(b) If (E ) - · is a family of ordered TVS 's-, K c. E a. a.c I a. - a. the positive ccne for each a., then· the product space

.· 1tE is. an ordered TVS for the cone K = 1tK , and nK a. a. a. is normal in nE if and only if K is normal in· E a.. . a a. for all a. t I.

Let E be ai: ordered TVS and K c. E a normal cono. u -- u Let Mc. E be a linear subspace. Then K N is a normal cone in H - u fl

for the subspace topology.- In addition if E is a vector l~ttice and u M s;. Eu a lattice ideal, then the canonical im_age of K in E /M is

a normal cone for the quotient topology.

Proof: See (23), (Chapter 2, ·S 1, Proposition 10).

Q) Definition: A series x. in a .loca.lly convex TVS is t1nconditionally L 1 i=l ·convergent if for any permutation p ·of the -indices., the series [ xp(i)

converg0s. 62

----Remark: Every series of real numbers is unconditionally co11veq:jent iff it is absolutely convergent.

Definition: I.et (x , f ) ""T. be a sep··rating biorthogonal .system in n n nc1 a locally convex -'I'VS E - Let - K GE - be the cone -associated with u· u

(x f ) Tneii (x ., :t ) is a basis for K if~every x c K n' n ncN" n n ncN

Let Eu be a locally convex, sequentialJ.y complete 'rVs,

~nd (x ,. f ) N a .separating biorthogonal .system such. that. (x , -f ) N _ n ·n he - n n ·nc

is a basi-s of the associa6 .1..t,ec_ ""1 cone K. Consider the followint; statements. co •.r {a) For any -'>. c f. {x) x. is unconditionally convergent. -!~' r .1 1 i=l CD (b) E,or any x c K· [ f. (x) x. is weakly uncondition2.lly ' i:::l J. J. converzept.

· (d) For x c K, p ;;:o f v c E : g < y < is_ bo:mc:ed. -x l" u x] .{<>' i( p ~1 For x c l. ,.~ i~ hcmeo.'Itor;:fri_ir. to a ctf be_, .,, x then. the following implications hold: (a)--7 (b), Cb) } (d_),

( ) -4 ( ) '-"''·-~(a,~ (.,''-\re.; C -:--1 a ' \ - / --J , ' .n I -1 ' - , o co If f.(x) x. is "ncondit:i.onally c0mrer1jent, L J. 1 .1

then it is weakly unconditionally convergent, hence the seq~ence of

parti&l sums is weakly _uncond.itiona.lly cauchy (see (4), Chapter !+ t 1).

(b) =::::> (d) Take x, -y <: K, y ~ x. 'I'hen trivialJ.y f. (y) < f. (x) ml, - 1 . for all i r. N. Since (" ) is a ba•c;is of K, __ y = t f., (y) x'" n , "'~n m:H 1 i::ol .J.

For f £ E', 63

f.(x) ·j·rcx.)I·~ Since aweukly unconditionally cauchy sequence in 1 . 1 .

· _a sequentially complete space is unconditionally ·convergeI;lt and in the

reals th~ latter is equivalent to absolute convergence, we get

CD .::--M -- If_(y )1 < f.(x)· f'Cx/ J <+ 00 for each· ·t t E ·where r· 1 I . f x i=l .. . ' is a positive real number de_pending upo_n f and x only. Hence Mf ,x .. for each x c K, Px - ( y:. Q ~. y ~· x J is .~eakly bounded, hence bounded in the inital topology by the Hackey theorem,

chapter, there e~ists a family of semi-norms (p ) ge.neratine.; u such a. o.cA

Since (x ) N is a basis of K for each there exists c > o. n nc a., · co N (d such that ( L f.(x) x.) < c for a.J.l n > N (c). Let a. i=n i l - a. CD Q) ·r· (:x) x· be any subseries of r: -f; (x) x .• .Choose i ·such that I: n .. n. i=l 1 .. 1 0 i=l 1 :!. q m n. > N (c) for -; > 1. • Th.en . for p, q _> i , g _::: [ f (x) x < ~- f.(x)x .• . 1 - ex. 0 0 n. n. . L :t i p 1 J. l=Na. (c) Since K is normal we have q (J) ,- .... ') .I. (x) x ...+' • ( XJ' x. ) < c: . 0 ~~a.(? n. n. )<- .p a. ( r;;-r ) ·l. 1 . 1 .. 1 . ~ a ,('.

Hence is cauchy with respect to each .a.., and

hence is cnuch;y with respect to tr..e oric;ihal topology u.

is sequentiaJly co~plete, f (x) x ·converges. But the sub.c;eries n. n. 1 l. Q) was arbifrary, hence 'oy ( l() , ( Ghn pter 1}, · S1, Page 60) , L f. (x) x. i=l. 1 1 is uncondit:i.onally convergent.

(e) ~ (d) 'l'his follO\v~ trivially since a· cub~, being the

product of compact sets is compact, hence bounded.

Assume holds. That P is h_ omeomorphic to Ca5 ·x· a· -cube follows from (6), Theor.1m 3. Q.E.D.

Remark: A cone K is called minihedral if for e:ny X; y- c K, _su1i G, y] exists. If (xn, fn)ncN iS an unconditional basis fbr the cone- K

associated 111ith it, then K is minihedral. .F'or if x, y c K, then

CD [ (f. (x) + f. (y))x. is unconditionally convergent by assumption, hence J. -- J. J. i=l CD r is also unconditicnally convergent because i=l

sup { fi (x), fi (y)] :S fi (x) + fi (y) and fi(x), fi (y) are > 0 for all

i c N, so

In this section we show (Thec1rem 7) that normality of a cone can

force existence of a basis fer the ccine. This is done in perfe_ct complete countably normed sp.'.:l.ces, thus cxt2nding a result known previously for a restricted class of 'Banach spaces sec: (21); Theorem 2. ·we ci.lso snow that ev.ery con€ associated.·1.-1ith a separating· hiorthoional syst.em is normal· in a perfect C.N.s.

For the definition of complete countably normed apace we refer the readm.~ to qhaptr~r I •. 65

Rema.rk: A C.N.S. - Eu. Gan be-made·into a metric space by. d(f, g)

llr-n~ p and the metric topology is identical with the p=lf:-1. 2P-- l+ _-g. np

original topology. Recall that by C.N.s. we mean complete countably

normed space.

'P. .· Theorem 2: If· a countabl-y normed space .LI satisfies the first axiom u

- of co1.mtab-ility, -then .:8' -is -ccmplete in the strong topoiogy - (3.

Proof: See (9), (Ch?pter 1, S _5).

Definition: A locally c0nvex TVS. is .perfect if all_ bounded sets are

relativol-y compact.

'l'heorem L~: Initial and \veak .convergence coincide in a perfect C.N.s.

Corollarv: A perfect C.N.s. is complete relative to weak ccnvergence. --·---~- --Proof: Follows as in (9), (Chapter 1, g 6). If E is a perfect c.:N. s. then weak and strong convergence u coincide in Et •

Pr_2_9,f: :for a proof see (9), (Chapter 1, £ 6).

CD Definition: A series x. in a TVS E -i-s ·weakly con.vergent if .L J. u i=l ro for any f- c E', -L 1-r

~Jlways converge to an element in the space. This is true onJy if the

space in ·weakly complete. 66

<» Proposition 5: (Orlicz) In a normed space B, a series E x i=l 1 converges weakly unconditionally if and onl7 if there exists f1 > 0

such that for any finite sequence of indices ~ < n2••• < l\c• l\ xnl + • • • + xnk Jl < M.

Proof: See (8), Theorem Page 241.

. Theorem 6: Let E be a perfect C.N.s. and (x , f ) N a u · n nm

separating biorthogonal system. Let K~ E be the associated cone. u .

Then if (l) K' - K' D E' K' the positive span of (f ), or ' n (2) For each (v ), (z ) where Q < v ·< z , (zn) bounded I "n n - "n - n implies (y ) bounded, then K is normal. n

Proof:

Q < v < z for n c N, and z ~ Q. We want to show that v ____, Q - "n - n n . "n and then apply Proposition 1 of this chapter to get K normal. Since I, ,

E is perfect, it is sufficient to show y ~ Q weakly. Since u · . n \ I zn~ Q, then fj(zn)~ fj(g) =0 _for all j c N. ·Also i

Q y z for all n gives v , z , z - y c K, n c N. This -< n -< n . "n n n n gives fj(zn - yn) 2: 0 or fj(zn) 2: fj(yn) for allk j, n c N•. So

fj(yn) ·---4 0 for each fixed j. For g c K', g = ~ u t , i=l 1 1 Observe that g(y ) ~ 0 as n ~ <».t since n k k • <2: a.. r. )(y ) .:$ IL (a. ri)(y >f .:$ < c' 1 1 1, n i=l 1 n

for suitably large n. Ct>, f c E', f = fl - t , fl, f c K'. By "By 2 2 t~' the above r1 (yn)~O and r2(°:n) ---?<>, hence (t1 - r2)(yn) f(y )-~ 0 as n ~<». Hence Y. 9 and K is normal. = n n ·--+ I'f (2) holds .we observe that• holds for any·g c ap (rn)• For f c E', f = l~m g~, gp c sp (fn) in the weak topology. This follows

since (f ) is separating, hence weakly total by [2), Page 51. n Hence f(yn) =Cr - gp + g~)(yn) .S \er - gp)(yn~ + \gp(yn)I • 'Given c ?' O, since gpC:r~>~o, there exists N(c, p) such

that for n ~ N(£, p), < c/2. Since z ---+ Q, by lsp < 1~' I n assumption (2), (y ) is bounc;led, and since E is perfect, n u

gp~f uniformly on bounrled sets, so there exists a k such

that for 13 ~ l{ t < c/2 for. aJ.1 Hf>nce ICr - gp)(yn>J · I Yn• Irey >I .S c/2 + c/2 :: c for n~ N(c, p), P ~ Therefore I l n Y • y ---) Q K n weakly and is normal.

Definition: (x , f ) Let . n n nc N be a separating biorthogonal system in a locally convex. TVS E u , and K SEu the associated cone. Then

(xn' rn)ncN is boundedly complete· over. K if for any sequence {a1) icN

of scalars such that a. > 0 for all a x ncN bounded 1 - 1 1J CD implies tllat..-:.-·r::;· a xi converges. \ i=l 1 Theorem ?: Let E be a perfect c.N.s. and {x , f ) Na separating u . n n~ biorthogonal system. Let KC E be the associated cone. .If · K is ,. - u normal then (xn' fn)ncN is an unconditional basis of K which is

also bounderlly complete on K.

Proof: To show (x , r ) N is boundedly complete, assume n n nc

ai ~ o, is bounded. Take Jf; N arbitrary but finite. fll ai xi) ncN n \ Then J f;; (1, n] for some n. Then g < E; a. xi :;S L; a x. from - icJ 1 i=l 1 1 the definition of K. Since K is normal by the previous Theoyem it follows from Proposition 3 of this chapter that there exists an equivalent countable family of semi-norms (qn)ncN such.that for 68

the defining family of norms for the C.N.s. there exists qk and N > O . k' n is the bound of· (k. ai x1 ) ncN with respect to qk. Hence taking M:: Nk ~ and applying Proposition 5

CD Cl> we r,et thar-t;;-rr for any t c E;. Since E' =\j' E' p=l p Cl> (9) (Chapter 1 S5," Page 45) , L l f(a. x.) I converges for each t c E_'. 1:::. 1 1 1

O> Hence fu ai xi converges weakly, and because Eu is perfect, [, ai xi 1=1 converges in the initial topology. Since the weak topology coincides with CD the initial topology (Theorem 4), L::. a x. converges in E ' and i=l 1 1 .u • since K is complete, ir converges to some element in K. Hence

is boundedly complete over. .K. \ To show (x , f ) N is a basis for K, observe that -Since n n nc ' (xn' f~)ncN is a biorthoeonal system, for x c K and using the definition of K, n Q _< ~ f. (x) x. x for all n. J:- 1 1< i=l Since K is normal, for n arbitrary and fixed arbitray p, there exists

Hence [ t. r1(x) xi} ncN is bounded in Eu' and from the above i=l 69

f. (x) x. converges. Since (f ) is neparating, f. ( x) x. ·- x. 1 1 n ncN J. l.

(p For if L f.(x) x. = y then for j fixed but arbitrary, i=l l. l. ()) . f .(y - x) = f. ( [ f. (x) x. - x ). = 0 J J . i~l 1 1 hence y x g or y ::::: x. This shoHs that (x f ) is a b9-sis - = n' n rn::N of K which is boundedly c'.'.inlplete over K. AJ.so by Theorem 1 of section 1, is DJ1 unconditional bas~ s sirice K is. norm:l1. (xn , f n ) nc N

Corollar,y.1: in a perfect C.N.S. }~ . Let KS Ti'_, be the <;.ssociated cone. If K is nortl-iq I and u 'U. ,,. "h' generating, then (x .L ) is a basis for .u n' n ncN u .

Proof: Since K is normal., (x f ,) is a basis for K, and n' n ncH since for .x l' x 2 c K. Hence

ro Q) (,r ) 'V' = L. r. Cy,) y f • •J ...,, ...... therefore x.'l. 2 =L ,_ c. :l i=l l. - i=l

0.J and the coeffic:ients are unique because for x = L.. 1\·x:i.' we have i;:::l (fl ( L> ( ,. Q f. (x) L- .I.. J] - y2) ·- f). )x. = and hence = fi (yl - y2) ·- [Ji i=l 1 - 1 l. ':!.

In this sectio:r:. we present some rE:sult;c; a'oc<..'.t ~ound~dJ.y complete and t;/:r_;e "P11 b'J.ses for cone!3.

Let E be a seDi-ref1cxive ·TVS and Cx , f ) ., a u · n n nn~ separating biorthog::.i::1al systerr. ·,:itr, the 8.fH:>ccir.tted cone K. I.f (x , f ) -, n n nc!J . b , er 1t; a ·'.':.!.S':!..S fer r_,:._· , t.r... ·"'. ·"·'· t ,... T ) is b0undedly cocplete over f'L.• - ' '""n' ··::i ncN 70

Assume a. > 0 scalars is a bounded subset l.

of E • Let c n ] • u be the circled convex cloSU:!:'e of [ f;: ai xi ncN 1 Then c is weakly compi"lct since E is semi-reflexive, see (18) u (Chapter 20). Hence ha.s a weak cluster point 5, £ [ [_ a. xi} m::N y, i=l J.

say. Clearly y c K, and we have f (y) > 0 for all n c N. Since n

(x f ) is a bas:!.s for K, there exists Cl unique sequence of n' n ncN CD ,., scalars a.. > 0 such that y a.. :-:: .• ...:i".nce<;!: v is weakly compact, -- r. 1 1 1 i=l

for any k c N, there exis ~,s a subsequence (nm(k)) such that

a. x. ) = 1 1 /.

Q) otherwise zero, hence a = ex. Hence a. x. converge~> k k. L J. 1 i=l and (x , f ) •r is boundedly complete over K. Q.E.D. n ~ m:1·

Definition: Let (x f ) be a biorthogonal system in. a locally n' n ndJ convex TVS Th.Em it is of type p if

(a) There exists a neighborhood V of zero such that x Iv n for e..11 n, e..nd n ( '01 ~ <-r x.1 [' is bo:mded. ti=l i J nc '•

Propo..3ition 7: Let. (xn' f·n ) ncN be a biorthogonal system in a loca.lly conYex TVS Eu sach that <·~,An' f''·n/ncN is :i. bc:,sis for th~ a.ssociated cone

K. If (xn, fn)ndl is of type P, then it is not boun'1edly complete

over K.

Proof: Assume (x • f ) · is of type P. 'I'h~:n t11ere exi.s ts a -·-·- n .n ndJ neir;hborhood V of ~ such th9.t x V for all- n. n I ?1

closed convex, circl~d, neighborhood of the origin such that W + W,s; V. n If (xn' f) ncN is bounctedly complete over K since ( '[_ x. }· N is t ·L::.:l 1 nc ro 'ii 'I bounded, L x. converges, say to x. Then xn =( .~ xi - x ) ·t i=l 1 , i=l n-1 ( x - L. x.) e::W+WSV for sufficiently large n. Contradiction! i=l l.' Hence (xn' fn)ncN is not baundedly complete over K.

Theorem 6 does not hold in a general C.N.s. For example take (I) ..--· E to be the closed hyperplane { x = (a ) c } : L u n 1 ai = 0} in X1, nili and let x - e e = Co, o, o, •.. , 1, o, a, ..• ). n n e n-l' where n

Then Cx ) M is a basis of 1'~ with the associated sequence of n nc" u rn coefficient functionals f (x) a . , x = Ca ) 111 by C28), Page 36l1-. n = L 1 n ncn i=l n The assoeintecl co'ne Ca } ,- is genera tir:g K = {x = n ne::N L. j_=l

but not normal sj_nce if it were, (xn_' fn)ne::N would tl1en be an unconditional

basis but (x , f ) , is not unconrtitional, see (28), Page 364. n n ncu11 BIBLIOGI?J\PHY

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