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 ΕΕΣ . Let ^ „ ( 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 ΕΕΣ . Let ^ „ ( E ) ...
Page 314
... weakly sequentially com- pact if and only if there exists a non - negative μ in ba ( S , E ) such that lim λ ( E ) = 0 μ ( E ) → 0 uniformly for λε Κ . PROOF . Let KC ba ( S , E ) 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 , E ) such that lim λ ( E ) = 0 μ ( E ) → 0 uniformly for λε Κ . PROOF . Let KC ba ( S , E ) 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 } , ƒ e 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 converging ...
... weakly sequentially compact if and only if it is bounded and the set { f } , ƒ e 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 converging ...
Contents
A Settheoretic Preliminaries | 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 Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ