From: "G. A. Edgar" Subject: Re: Duals of non-reflexive Banach spaces. Date: 11 Nov 2000 10:30:04 -0600 Newsgroups: sci.math.research > Could someone, more Banach-space aware than I, refer me to a textbook > (preferably) or paper answering these obvious questions, and giving examples > where they exist: As a textbook you should try "Classical Banach Spaces" by Lindenstrauss & Tzafriri (volume I). -- Gerald A. Edgar edgar@math.ohio-state.edu ============================================================================== From: Adrian Duma Subject: Some other examples Date: 11 Nov 2000 14:05:02 -0600 Newsgroups: sci.math.research Recently Jonathan LF King raised the following problems: "Q1: Is there a B-space which is the dual-space of two non-isomorphic B-spaces?" Answer: Yes, there is a nonseparable Banach space X whose dual X* is isometrically isomorphic to (C[0,1])*. "Q2: Is there a B-spaces which is the dual of no B-space?" Answer: Yes, say L^1[0,1]. In order to check these answers, please see the classical monograph of H.Elton Lacey "The isometric theory of classical Banach spaces" (1974). Hope this helps. With best regards, Adrian Duma. ============================================================================== From: Stephen Montgomery-Smith Subject: Re: Schauder bases (was Re: Duals of non-reflexive Banach spaces) Date: 12 Nov 2000 01:30:02 -0600 Newsgroups: sci.math.research Ronald Bruck wrote: > > A colleague recently asked me for a reference on Schauder bases in > concrete spaces--that is, he actually wants SPECIFIC bases in L^p(R), > C[0,1], etc. I tossed back "Classical Banach Spaces" as a good place to > look, because I was **sure** I remembered seeing examples there. > > But I seem to have misremembered; he took the book out of the library > and couldn't find anything. I've searched through all my books on > Banach spaces, and can't find anything. I **know** I've seen these in a > standard reference. Does anyone have any suggestions for where to look? > I remember seeing something like this in the book by Joe Diestel titled something like "Sequences and Series in Banach Spaces." Actually, I think I can remember them. For L_p([0,1]) take functions e_{n,k} n=0,1,2,3,... , 0<=k<2^n e_{n,k}(t) = 1 if k < 2^n t < k+1/2 e_{n,k}(t) = -1 if k+1/2 < 2^n t < k+1 e_{n,k}(t) = 0 otherwise or something like that. Throw in the function 1, and you have the Schauder basis for L_p([0,1]), 1<=p1 it is an unconditional basis. As for C([0,1]), take the antiderivatives of C([0,1]). -- Stephen Montgomery-Smith stephen@math.missouri.edu http://www.math.missouri.edu/~stephen ============================================================================== From: Adrian Duma Subject: Re:Schauder bases Date: 12 Nov 2000 20:05:03 -0600 Newsgroups: sci.math.research Recently, Ronald Bruck wrote: "A colleague recently asked me for a reference on Schauder bases in concrete spaces--that is, he actually wants SPECIFIC bases in L^p(R), C[0,1], etc." "Does anyone have any suggestions for where to look?" In my opinion, one of the best monographs concerning this subject is Ivan Singer's one, namely "Bases in Banach Spaces", volume I, Springer Verlag, 1970, and volume II, Springer too, 1981. Actually, I don't know nothing about the third part of this monograph (1989?, Springer?), which was (in 1981) promised to appear. This (hypothetical?) third part should be concerned exactly with concrete Schauder bases, but you may also take a look to volumes I and II. Hope this helps. With best regards, Adrian Duma. ============================================================================== From: Yossi Lonke Subject: Read, and ya shall find Date: 13 Nov 2000 22:05:02 -0600 Newsgroups: sci.math.research --But I seem to have misremembered; he took the book out of the library --and couldn't find anything. What do you mean by "There are no examples of Schauder bases in Lindenstrauss-Tzafriri" ?? On page 3, there is the explicit construction of the Haar basis, which is a Schauder basis in L^p(0,1) for all 1<=p < inifinity. A few lines below, on the same page, there is the "Schauder system", obtained by integrating the Haar system. That gives an explicit Shcauder basis in C(0,1), answering the question of your friend (and he only needed to read up to page 3!!) On page 4, another specific basis is presented, this time for the disc algebra -- namely the Franklin system. Another good source is the book "Banach spaces for Analysts" by Wojtaszczyk. on p. 39 the Haar basis is constructed. on p. 40 -- the Faber-Schauder basis. Yossi Lonke Ronald Bruck wrote: > In article <111120000902248618%edgar@math.ohio-state.edu.nospam>, "G. > A. Edgar" wrote: > > :> Could someone, more Banach-space aware than I, refer me to a textbook > :> (preferably) or paper answering these obvious questions, and giving > :> examples > :> where they exist: > : > :As a textbook you should try "Classical Banach Spaces" by Lindenstrauss > :& Tzafriri (volume I). > > A colleague recently asked me for a reference on Schauder bases in > concrete spaces--that is, he actually wants SPECIFIC bases in L^p(R), > C[0,1], etc. I tossed back "Classical Banach Spaces" as a good place to > look, because I was **sure** I remembered seeing examples there. > > But I seem to have misremembered; he took the book out of the library > and couldn't find anything. I've searched through all my books on > Banach spaces, and can't find anything. I **know** I've seen these in a > standard reference. Does anyone have any suggestions for where to look? > > --Ron Bruck > > -- > Due to University fiscal constraints, .sigs may not be exceed one > line. -- ************************************************* Dr. Yossi Lonke Mathematics Department Case Western Reserve University 10900 Euclid Avenue Cleveland, Ohio 44106 216 368-5423 http://www.cwru.edu/artsci/math/lonke/home.html *************************************************