Let me see if I have this correct -- You AVt, Graeme, Diestel, Dunford and Schwartz, John B. Conway, Edmunds and Evans, Halmos, Akhiezer and Glazman, and Lindenstrauss and Tzafriri think that banach space is simply a linear vector space??? I'd say you all don't have a clue. Actually you all don't have a clue whether I say so or not. Yes I am asserting 'the earth is locally flat'. Otherwise you wouldn't have a local diffusion for an exp map when solving problems using stochastic differential geometry....AVt, Here's a special treat for you. Are there any prime numbers, per se, in banach space. No. Only primes as factors. Why? Every number must have 2 non-unity factors for the field to be cyclotomic. Of course, one and only one arbitary scaling factor is permitted (which can be any real number). As a side note, I think Fermat knew that when he scribbled in his margin. Newton Edit: Let's refer to Banach as the space without Hilbert space. If we mean hilbert space then we'll be explicit. N,That is one of your real gems My tribe and me will discuss it with our trustworthiness shaman in the dark Bavarian woods at next Full Moon, so about 2 weeks later i will prepared for the seance with your ancient namesake PS: please go on.

QuoteOriginally posted by: NLet me see if I have this correct --You AVt, Graeme, Diestel, Dunford and Schwartz, John B. Conway, Edmunds and Evans, Halmos, Akhiezer and Glazman, and Lindenstrauss and Tzafriri think that banach space is simply a linear vector space???No. We are asserting that a Banach space by definition is a linear vector space with a norm under which it is complete. Given that this is what I wrote in my post on Sunday, you surely will see that it is difficult to conduct a meaningful dialogue with somebody who cannot even read. Further, please note that this is a DEFINITION, which is accepted by most people on this planet. As such, you cannot argue with this. If you want to DEFINE a different type of space, you need to 1. propose a name, one not already in use for anything else (in particular, not "Banach space") 2. give the defining properties. I have not proposed any results or provided counter-examples to proposed results on Banach spaces, and I don't think anybody else has on this thread. I and others have only given the definition of a Banach space, and have given a couple of simple examples of spaces which are Banach spaces i.e. a) are linear topological spaces b) have a norm c) as topological spaces, are complete under this norm. And nothing else: no more, no less.This is the usual approach in mathematics. You are welcome to subscribe to it, but if not, you need to find a different arena of discourse.

QuoteOriginally posted by: nsandeI think I got it.N's real name has to be Bogdanov.Regards,NiclasNo. There are still genuine academics who believe that the Bs are not necessarily B.S.

The actual common definition of Banach Space is a Metric space with a norm. Period. The space doesn't need to be linear or complete.Hilbert space is a Metric Space with a norm and has the additional inner product property.That's it. Many derive a few more properties as I did (usually like the triangular inequality, etc.)

QuoteOriginally posted by: NThe actual common definition of Banach Space is a Metric space with a norm. Period. The space doesn't need to be linear or complete.Hilbert space is a Metric Space with a norm and has the additional inner product property.That's it. Many derive a few more properties as I did (usually like the triangular inequality, etc.)Newton, a banach space is, by definition, complete. It is a complete space you bonehead.

C3--Check the definition of complete.How can a space be complete if it's missing the primes???? Everything is also stuck in fixed orbits determined by the p-norm. Who's the bonehead?Hey, It's not my fault that there's always quadrature when the p-norm isn't two.

QuoteOriginally posted by: NC3--Check the definition of complete.How can a space be complete if it's missing the primes???? Everything is also stuck in fixed orbits determined by the p-norm. Who's the bonehead?Hey, It's not my fault that there's always quadrature when the p-norm isn't two.Fine Newton, you deranged goofball. Please do me one favor and show me a banach space that has divergent subsequences. in other words, tell me about a well know space that has a cauchy sequence in it blow up.

That's easy.The divergent sequence of partial sums of the harmonic series. Any partial sums of an elliptic curve series from two orthogonal spinners from any p-norm <>2 Banach space. Remember stuff in banach space all reside in orbits.

Better yet.Any series where you need to apply a renormalization group to get convergence. The renormalization is a sinc function which rotates global coordinates to match the rotation of the problem spinnor.Notice that Spinnors (Clifford Algebra) never have this problem.

golfball, help me with your comment about the 'finite geometries' thing... what are you getting at here?why does a banach space have integers? I did not see that as part of the definition either. And without integers, why even mention primes?but back to 'finite geometries'... you know, something like Q(sqrt(17)) is finitely generated, but not cyclotomic, has lots of primes, integers even, but alas... is not a Banach space! I know I'm not a genius but I don't see where the fucking Clifford Algebra is hiding? or did I forget to include the d'alembertian operator in the definition?your student,kr