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By E. M. Friedlander, M. R. Stein

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2 ( 0 ) Let T b e a s e t a n d g~ := {x E 1KTI x is bounded}. ll~:f~162 IZ T and ~IR+, x, ~suplx(t)I tET is a norm, called the s u p r e m u m norm. t ~ endowed with this norm is a Banach space. It is separable iff T is finite. We set f ~ := f~(iN). ll~ is a norm. 1 Normed Spaces 13 for every m, n E IN, with m > n~, n > n~. Hence, given t C T , the sequence (xn(t))ne~ is a Cauchy sequence (and so a convergent sequence) in ]K. Define x'T >IK, tD > limxn(t). u-+co Then for all n E IN, n > n~, sup ixn(t) - x(t)l < c.

First assume that (x~)~el is summable and let x be its sum. Then there is a J E ~ / ( I ) with Xe--X tEK rl ~ -~ for every K E ~3S(I),J C K . Take K E ~ f ( I \ J ) . finite subsets of I containing J , hence ~EK x~ XL_X eEJUK Then J and J U K X -- ~ - ' ~ X t are c c <~+~=~. ~EJ Now assume that E is complete and that the condition described above is fulfilled.

4 ( 0 ) Given n E IN U {O, o c } , we define ( 0 ) (Mi,~ow~i'~ Theorem, ~Sg~). Norm~ of finite-dimensional vector space are equivalent. Given n c IN, let p be an arbitrary norm on IK n and I1" II the Euclidean norm on IKn . Furthermore, let el, e 2 , . . , en be the standard basis vectors of IK n and put a := p(ek) 2 > O. k--1 Given x := (xk)ke~. 3). Hence p is continuous. Define K := {x E IK'~ ] IIx]] = 1}, := inf{p(x) l x e K } . - ~11~11. I. 1 Normed Spaces 33 By the Bolzano-Weierstrass Theorem, K is compact.

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Algebraic K-Theory by E. M. Friedlander, M. R. Stein

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