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 97
... bounded variation if ( μ , S ) < ∞ , and it is of bounded variation on a set E in Σ if v ( μ , E ) < ∞ . 5 LEMMA . If a real or complex valued additive set function defined on a field Σ of subsets of a set S is bounded , it is of bounded ...
... bounded variation if ( μ , S ) < ∞ , and it is of bounded variation on a set E in Σ if v ( μ , E ) < ∞ . 5 LEMMA . If a real or complex valued additive set function defined on a field Σ of subsets of a set S is bounded , it is of bounded ...
Page 345
... bounded and the set { f ( P ) } , ƒ € 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 fin A ( D ) ) if and only if ...
... bounded and the set { f ( P ) } , ƒ € 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 fin A ( D ) ) if and only if ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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 compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ