Linear Operators: General theory |
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Page 54
... bounded sets into bounded sets . - > Y PROOF . Let X , Y be F - spaces , let T : X → be linear and contin- uous , and let BCX be bounded . For every neighborhood V of the zero in Y , there is a neighborhood U of zero in X such that T ...
... bounded sets into bounded sets . - > Y PROOF . Let X , Y be F - spaces , let T : X → be linear and contin- uous , and let BCX be bounded . For every neighborhood V of the zero in Y , there is a neighborhood U of zero in X such that T ...
Page 345
... 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 to f in A ( D ) ) if and only if it ...
... 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 to f in A ( D ) ) if and only if it ...
Page 436
... bounded ( II.1.7 ) set , then X must be finite dimensional . 6 Let X be a B - space , and X1 a subspace of X. Show ... bounded if and only if f ( A ) is a bounded set of scalars for each ƒ € l . 8 Let X be a B - space . Show that a ...
... bounded ( II.1.7 ) set , then X must be finite dimensional . 6 Let X be a B - space , and X1 a subspace of X. Show ... bounded if and only if f ( A ) is a bounded set of scalars for each ƒ € l . 8 Let X be a B - space . Show that a ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ