Linear Operators: General theory |
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Page 54
... bounded sets into bounded sets . PROOF . Let X , Y be F - spaces , let T : X → y 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 . PROOF . Let X , Y be F - spaces , let T : X → y 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 v ( 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 variation if v ( 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 ...
Page 345
... bounded and the set { f ( P ) } , ƒ e A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in 4 ( D ) is a weak Cauchy sequence ( a sequence converging weakly to ƒ in A ( D ) ) if and only ...
... bounded and the set { f ( P ) } , ƒ e A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in 4 ( D ) is a weak Cauchy sequence ( a sequence converging weakly to ƒ in A ( D ) ) if and only ...
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