From: ullrich@math.okstate.edu (David C. Ullrich) Subject: Re: Banach space question Date: Thu, 04 May 2000 15:22:03 GMT Newsgroups: sci.math Summary: modified norms on Banach spaces On Wed, 03 May 2000 21:37:25 GMT, K wrote: >Let X be a real Banach space with basis (x_j); i.e. >for all x in X there is a unique sequence (a_j) of real numbers with > >x = sum(a_j*x_j,j=1..infinity). > >Define P(n):X->X by P(n)(x) = sum(a_j*x_j,j=1..n). > >Define p:X->R by p(x) = sup{||P(n)(x)||,n>=1). > >It can be shown that p is a norm on X. > >Show that (X,p) is complete. > >Any takers? This is a very standard result in Banach space theory. I could give a hint or try to find a reference, but it's more fun to see whether Pertti gets it. Ok. If you look in Wojtaszczyk "Banach Spaces for Analysts" you find a proof that the P_n are _uniformly_ bounded (ie sup ||P_n|| < infinity; note that the fact that P_n is bounded is not quite as obvious as you might think). The completeness of p follows easily from this. >Sent via Deja.com http://www.deja.com/ >Before you buy. From ullrich@math.okstate.edu Thu May 18 23:08:56 CDT 2000 Article: 315554 of sci.math Path: news.math.niu.edu!husk.cso.niu.edu!vixen.cso.uiuc.edu!howland.erols.net!pants.skycache.com.MISMATCH!triton.skycache.com!128.230.129.106!news.maxwell.syr.edu!nntp2.deja.com!nnrp1.deja.com!not-for-mail From: ullrich@math.okstate.edu Newsgroups: sci.math Subject: Re: Banach space question Date: Mon, 15 May 2000 18:50:58 GMT Organization: Deja.com - Before you buy. Lines: 89 Message-ID: <8fpgub$qdg$1@nnrp1.deja.com> References: <8eq66j$vdl$1@nnrp1.deja.com> <8f1euq$td5$1@nnrp1.deja.com> <8fnk7i$i5v$1@slb7.atl.mindspring.net> NNTP-Posting-Host: 139.78.115.111 X-Article-Creation-Date: Mon May 15 18:50:58 2000 GMT X-Http-User-Agent: Mozilla/4.5 [en] (Win95; U) X-Http-Proxy: 1.0 x41.deja.com:80 (Squid/1.1.22) for client 139.78.115.111 X-MyDeja-Info: XMYDJUIDullrich Xref: news.math.niu.edu sci.math:315554 In article <8fnk7i$i5v$1@slb7.atl.mindspring.net>, "Daniel Giaimo" wrote: > > wrote in message > news:8f1euq$td5$1@nnrp1.deja.com... > > Sorry if this appears twice - the local news server has been > > very confused (almost as confused as the people running it...) > > > > In article <8eq66j$vdl$1@nnrp1.deja.com>, > > K wrote: > > > Let X be a real Banach space with basis (x_j); i.e. > > > for all x in X there is a unique sequence (a_j) of real numbers with > > > > > > x = sum(a_j*x_j,j=1..infinity). > > > > > > Define P(n):X->X by P(n)(x) = sum(a_j*x_j,j=1..n). > > > > > > Define p:X->R by p(x) = sup{||P(n)(x)||,n>=1). > > > > > > It can be shown that p is a norm on X. > > > > > > Show that (X,p) is complete. > > > > This is a standard result. If you look in Wojtaszczyk > > "Banach Spaces for Analysts" you can find a proof that > > > > sup(||P(n)||) < infinity, > > > > from which the result follows easily. (The finiteness of that > > sup implies that p is equivalent to the original norm). > > > > There's a curious aspect to this problem - there's an > > extremely trivial proof that sup(||P(n)||) < infinity, which > > happens to be wrong wrong wrong: We know that P(n)(x) -> x > > in norm, for all x in X. Hence ||P(n)(x)|| is bounded, for > > all x in X, and it then follows from the Uniform Boundedness > > Principle (aka the Banach-Seinhaus theorem) that ||P(n)|| is > > bounded. > > Hmmm... > > It doesn't seem immediately apparent that the P(n)'s are > _bounded_ linear maps, so the Uniform Boundedness Principle > doesn't obviously apply. Am I right? Yup. Always seemed very strange to me - an argument that gives a bound on a bunch of numbers if they happen to be finite should also show they're finite, but this one doesn't. Like: Suppose that f:R->R satisfies | (f(x-h) - 2*f((x) + f(x+h)) | <= |h| for all x, h. Does this imply f is continuous? No. But if we assume in addition that f is continuous then a specific estimate on the modulus of continuity follows! (|f(x+h) - f(x)| <= c*|h log(1/h)| for small h). > What do I win? Am I > a great mathematician now? You get to start posting messages about finding an error of Ullrich. I'll say a few times that I _said_ it was wrong. It's important that you ignore it when I say this, and continue to gloat about having found the error. If you want to be truly great any time I say anything for the rest of my life you will post a link to a web page claiming that I have never even acknowledged the error, because I don't like to take opportunities to institiute cognitive blah blah blah. This is why it's important to pay no attention when I _do_ acknowledge the blunder. > > If anyone can say what the error is in the non-proof > > above he gets to post 10,000 messages saying he was the > > first to spot Ullrich's blunder. The fact that explicitly > > _said_ it was wrong shouldn't stop you - I said something > > wrong, and if you're the first to point it out that will > > make you a great mathematician. (No fair looking in > > Wojtaszczyk first - he explicitly says that the hard part > > is showing something that I totally omitted above.) > > > > Sent via Deja.com http://www.deja.com/ Before you buy.