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Page 54
... mapping of one F - space into another is continuous if and only if it maps 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 ...
... mapping of one F - space into another is continuous if and only if it maps 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 ...
Page 55
... Mapping Principle The interior mapping principle is stated in the following theorem . 1 THEOREM . Under a continuous linear map of one F - space onto all of another , the image of every open set is open . y PROOF . Let X , Y be F ...
... Mapping Principle The interior mapping principle is stated in the following theorem . 1 THEOREM . Under a continuous linear map of one F - space onto all of another , the image of every open set is open . y PROOF . Let X , Y be F ...
Page 478
... mapping from * to X * defined by T * y * = y * T . 2 LEMMA . The mapping T → T * is an isometric isomorphism of B ( X , Y ) into B ( Y * , X * ) . PROOF . The linear functional y * T is continuous ( I.4.17 ) , and hence T * y * X * . The ...
... mapping from * to X * defined by T * y * = y * T . 2 LEMMA . The mapping T → T * is an isometric isomorphism of B ( X , Y ) into B ( Y * , X * ) . PROOF . The linear functional y * T is continuous ( I.4.17 ) , and hence T * y * X * . The ...
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