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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 ...
... 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 ...
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 513
... mapping T → T * of B ( § ) into itself is continuous with either the uniform or weak operator topology . By considering the sequence { 4 } defined in Exercise 11 , show that this mapping is not continuous in the strong operator ...
... mapping T → T * of B ( § ) into itself is continuous with either the uniform or weak operator topology . By considering the sequence { 4 } defined in Exercise 11 , show that this mapping is not continuous in the strong operator ...
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