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Bases and Cones in Locally Convex Spaces

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Bases and Cones in Locally Convex Spaces BASES AND CONES IN l,OCALLY CONVEX SPACES By 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<ble Banach space have a Schaudr:r bc-~sis? Ha.ny people have tried in vain to solve this problem and I sup-oose 1.;hat a con.:si.derable part of bc:tsia theory has been motivated to some cx:tent by this probJ.,:m, 3 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. (c) Let M> E, M /. 0. The smallest circled set containing M is called the circled hull of M.
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