Linear Operators: General theory |
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Page 20
... complete metric space is com- plete . A complete subspace of a metric space is closed . 8 LEMMA . A mapping f : X → Y between metric spaces is contin- uous at the point x if and only if { f ( xn ) } converges to f ( x ) , whenever { x } ...
... complete metric space is com- plete . A complete subspace of a metric space is closed . 8 LEMMA . A mapping f : X → Y between metric spaces is contin- uous at the point x if and only if { f ( xn ) } converges to f ( x ) , whenever { x } ...
Page 311
... complete . If S is a topological space , the rba ( S ) is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of B ( S ) . According to Theorems 6.18 and 6.20 there is a compact Hausdorff space S1 such that B ( S , E ) ...
... complete . If S is a topological space , the rba ( S ) is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of B ( S ) . According to Theorems 6.18 and 6.20 there is a compact Hausdorff space S1 such that B ( S , E ) ...
Page 840
... Complete and o - complete lattice , ( 43 ) Complete metric space , compact , 1.6.15 ( 22 ) definition , 1.6.5 ( 19 ) properties , 1.6.7 ( 20 ) , 1.6.9 ( 20 ) Complete normed linear space . ( See B - space ) Complete orthonormal set , in ...
... Complete and o - complete lattice , ( 43 ) Complete metric space , compact , 1.6.15 ( 22 ) definition , 1.6.5 ( 19 ) properties , 1.6.7 ( 20 ) , 1.6.9 ( 20 ) Complete normed linear space . ( See B - space ) Complete orthonormal set , in ...
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 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ