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Page 259
... conditionally compact ( I.6.15 ) , it will be sufficient and convenient for our present purposes to state conditions ... compact set . Conversely , if lima Uax = x uniformly for x in a bounded set K , and if , in addition , Ua ( { x || x ...
... conditionally compact ( I.6.15 ) , it will be sufficient and convenient for our present purposes to state conditions ... compact set . Conversely , if lima Uax = x uniformly for x in a bounded set K , and if , in addition , Ua ( { x || x ...
Page 266
... conditionally compact if and only if for every & > 0 there is a finite collection { E , ... , En } of sets with ... compact , then a set in C ( S ) is conditionally compact if and only if it is bounded and equicontinuous . PROOF . Let K ...
... conditionally compact if and only if for every & > 0 there is a finite collection { E , ... , En } of sets with ... compact , then a set in C ( S ) is conditionally compact if and only if it is bounded and equicontinuous . PROOF . Let K ...
Page 267
... compact metric space and K be a bounded set in C ( S ) . Then K is conditionally compact if and only if for every ε > 0 there is a 80 such that sup f ( s ) —f ( t ) | < ɛ , fεK o ( s , t ) < 8 . PROOF . If the condition is satisfied ...
... compact metric space and K be a bounded set in C ( S ) . Then K is conditionally compact if and only if for every ε > 0 there is a 80 such that sup f ( s ) —f ( t ) | < ɛ , fεK o ( s , t ) < 8 . PROOF . If the condition is satisfied ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ