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Page 294
... sequentially compact then lim f ( s ) μ ( ds ) = 0 , μ ( E ) → 0 uniformly for fin K. If μ ( S ) < ∞ then conversely this condition is suffi- cient for a bounded set K to be weakly sequentially compact . PROOF . Let K be a weakly ...
... sequentially compact then lim f ( s ) μ ( ds ) = 0 , μ ( E ) → 0 uniformly for fin K. If μ ( S ) < ∞ then conversely this condition is suffi- cient for a bounded set K to be weakly sequentially compact . PROOF . Let K be a weakly ...
Page 314
... sequentially com- pact if and only if there exists a non - negative μ in ba ( S , Σ ) such that uniformly for λεΚ . - lim λ ( E ) **** 0 μ ( E ) → 0 PROOF . Let KC ba ( S , 2 ) be weakly sequentially compact and let V UT be the ...
... sequentially com- pact if and only if there exists a non - negative μ in ba ( S , Σ ) such that uniformly for λεΚ . - lim λ ( E ) **** 0 μ ( E ) → 0 PROOF . Let KC ba ( S , 2 ) be weakly sequentially compact and let V UT be the ...
Page 343
... sequentially compact and we have fn ( s ) → f ( s ) for s in I. 31 A bounded subset K of AC ( I ) is sequentially weakly compact if and only if for each ɛ > 0 there is a d > 0 such that Σ n Σst 8 implies Σf ( s ) -f ( t ; ) | < ε i = 1 ...
... sequentially compact and we have fn ( s ) → f ( s ) for s in I. 31 A bounded subset K of AC ( I ) is sequentially weakly compact if and only if for each ɛ > 0 there is a d > 0 such that Σ n Σst 8 implies Σf ( s ) -f ( t ; ) | < ε i = 1 ...
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