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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 lim , Uxx 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 lim , Uxx 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 ...
... 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 ...
Page 267
... compact . Conversely , let K be conditionally compact and suppose the con- dition is not satisfied . Then there is an ε > 0 and sequences { n } , { tn } S , { fn } CK with fn ( Sn ) —fn ( tn ) > ε , Q ( 8n , tn ) 0. Since S and K are ...
... compact . Conversely , let K be conditionally compact and suppose the con- dition is not satisfied . Then there is an ε > 0 and sequences { n } , { tn } S , { fn } CK with fn ( Sn ) —fn ( tn ) > ε , Q ( 8n , tn ) 0. Since S and K are ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ