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 ( U ) ...
... 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 ( U ) ...
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 ( 4 ) is a bounded set of scalars for each ƒ € I. 8 Let X be a B - space . Show that a subset ...
... 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 ( 4 ) is a bounded set of scalars for each ƒ € I. 8 Let X be a B - space . Show that a subset ...
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