# Banach Spaces by Jeffrey M. Lemm By Jeffrey M. Lemm

Hardbound.

Similar topology books

Modern Geometry: Introduction to Homology Theory Pt. 3: Methods and Applications

During the last fifteen years, the geometrical and topological equipment of the idea of manifolds have assumed a vital function within the such a lot complex parts of natural and utilized arithmetic in addition to theoretical physics. the 3 volumes of "Modern Geometry - equipment and functions" comprise a concrete exposition of those equipment including their major functions in arithmetic and physics.

Borel Liftings of Borel Sets: Some Decidable and Undecidable Statements

One of many goals of this paintings is to enquire a few usual homes of Borel units that are undecidable in $ZFC$. The authors' start line is the next effortless, notwithstanding non-trivial end result: examine $X \subset 2omega\times2omega$, set $Y=\pi(X)$, the place $\pi$ denotes the canonical projection of $2omega\times2omega$ onto the 1st issue, and consider that $(\star)$ : ""Any compact subset of $Y$ is the projection of a few compact subset of $X$"".

Example text

Norm of E to F is a norm on F is The restriction of the (the induced n o r m on F ); F endowed with this norm is called subspace o f E . Normed spaces over IK are called real n o r m e d spaces if ]K = ]R and c o m p l e x n o r m e d spaces if ]K = (~. I f they are complete, we call them real (rasp. c o m p l e x ) B a n a c h spaces. Given a metric space T , we define for each t C T and ce > O, Us(t) : - u T ( t ) : = {s c T Id(s, t) < c~}, where d denotes the metric of T . Remark. 1. In 1908 M.

5 ( 0 ) 39 Let E be a normed space, A a subset of E , and F the vector subspace of E generated by A . Then F is the smallest closed vector subspace of E constaining A . It is called the closed vector subspace o f E generated by A . If A is countable, then F , endowed with the induced norm, is separable. 4, F is a vector subspace of E and it is clear that it is the smallest closed vector subspace of E containing A. Assume now that A is countable and let B be the set of linear combinations of elements of A with coefficients in Q (resp.

Given n E IN, put 1 y n : = nq'x,~'[ ) x n Then, given n C IN P(Yn) ~- p(Xn) > 1, nq(xn) q(v~) = q(~) _ 1 Hence lim q(Yn) = 0 n---+oo and so, by c), lim P(Yn) = O, n--+oo which is a contradiction. Hence there is some c~ > 0 with p _< a q . Interchanging the roles of p and q, it follows t h a t p and q are equivalent. I 32 1. Banach Spaces Remark. For a substantial part of the theory of normed spaces (which includes the first four chapters of this book) only the topology generated by the norm, and not the norm itself is significant. 