Linear Operators: General theory |
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Page 294
... weakly sequentially compact , a subsequence of { f } con- verges weakly , and it may be assumed without loss of generality that { f } itself converges weakly . By Theorem 7 , { SEfn ( s ) u ( ds ) } converges for each Ε Ε Σ . Let λn ( E ) ...
... weakly sequentially compact , a subsequence of { f } con- verges weakly , and it may be assumed without loss of generality that { f } itself converges weakly . By Theorem 7 , { SEfn ( s ) u ( ds ) } converges for each Ε Ε Σ . Let λn ( E ) ...
Page 314
... weakly 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 ...
... weakly 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 345
... weakly sequentially compact if and only if it is bounded and the set { f ( P ) } , fe A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in A ( D ) is a weak Cauchy sequence ( a sequence ...
... weakly sequentially compact if and only if it is bounded and the set { f ( P ) } , fe A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in A ( D ) is a weak Cauchy sequence ( a sequence ...
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