 By Jeffrey M. Lemm

ISBN-10: 0444507493

ISBN-13: 9780444507495

ISBN-10: 0444507582

ISBN-13: 9780444507587

ISBN-10: 1865843830

ISBN-13: 9781865843834

Hardbound.

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Example text

We d~fn~ E ~ : - { x e E l l x l <1}. E # is called the unit ball o f E . Two norms p, q on a vector space are said to be equivalent if 1 - p <_ q <_ c~p o~ for some ~ > O. ll. Given E a normed space, the map E • E ~ ~+, (x,y) , ) l l ~ - yll is a metric on E , called the c a n o n i c a l m e t r i c o f E . Unless otherwise specified, we take every normed space to be endowed with its canonical metric. Its topology is called the n o r m topology. I f this metric is complete, then we say that the norm is c o m p l e t e or that E is complete.

Given c > 0 there is an n E E IN with IlZm -- x~ll~ < for m , n E IN with m > n ~ , n > n . We deduce t h a t for every t E T , (x~(t))~E~ is a Cauchy sequence. Define x'T >IK, t, > limx=(t). n - - + (x) Then 1 p _ n~. P(T) and lim xn = x . n--+ (x) gP(T) is thus a Banach space. If T is infinite then the set A - {x E ]K(T) I t e T ~ rex(t), imx(t) E~} is a dense set of gP(T) and Card A - Card T . We have 1 i_ lies - etll = 2~ 1 -- 16 1.

TCB Given t E B , choose st E A M Ut. T" in the topology of pointwise convergence in A. Take y E H and s C T . Then there is a t E B with x C Ut. Thus II~(s) - y(s)ll < II~(s) - x(t)ll + Ilk(t) - ~ ( ~ d l l + c + II~(s,) - y ( s , ) l I + Ily(s,) - y(t)ll + Ily(t) - y(~)ll < 5 g = c. T" of pointwise convergence in A is finer than the topology on ~ of uniform convergence on T . Since the reverse relation is trivial, the two topologies on ~ coincide. 16 ( 0 ) I (Ascoli, 1883; Arzels 1889) Let T be a compact space.