Linear Operators: General theory |
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Page 22
... sequentially compact . Conversely , suppose that A is sequentially compact and closed . It will first be shown that A is separable . Let po be an arbitrary point of A , and let do sup ( Po p ) . The number do is finite , for if PEA e ...
... sequentially compact . Conversely , suppose that A is sequentially compact and closed . It will first be shown that A is separable . Let po be an arbitrary point of A , and let do sup ( Po p ) . The number do is finite , for if PEA e ...
Page 314
... sequentially com- pact if and only if there exists a non - negative μ in ba ( S , E ) such that 12 uniformly for λεΚ . lim 2 ( E ) = 0 μ ( E ) -0 PROOF . Let KC ba ( S , E ) be weakly sequentially compact and let ' UT be the isometric ...
... sequentially com- pact if and only if there exists a non - negative μ in ba ( S , E ) such that 12 uniformly for λεΚ . lim 2 ( E ) = 0 μ ( E ) -0 PROOF . Let KC ba ( S , E ) be weakly sequentially compact and let ' UT be the isometric ...
Page 511
... compact in the weak operator topology if and only if it is closed in the weak operator topology and the weak closure ... sequentially compact in the weak operator topology , its weak operator closure is compact in the weak operator ...
... compact in the weak operator topology if and only if it is closed in the weak operator topology and the weak closure ... sequentially compact in the weak operator topology , its weak operator closure is compact in the weak operator ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ