By Jeffrey M. Lemm

Hardbound.

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**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.