Linear Operators: General theory |
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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 . PROOF . Let X , Y be F - spaces , let T : X → y be linear and contin- uous , and let BC 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 . PROOF . Let X , Y be F - spaces , let T : X → y be linear and contin- uous , and let BC 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 . PROOF . Let X , Y be F - spaces ...
... 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 . PROOF . Let X , Y be F - spaces ...
Page 493
... mapping which sends a into u ( · ) x * , then it follows that for each fixed fe C ( S ) the mapping x * → > [ st ( s ) μ ( ds ) x * is continuous in the X topology of X * and therefore ( V.3.9 ) is generated by some element x , e X ...
... mapping which sends a into u ( · ) x * , then it follows that for each fixed fe C ( S ) the mapping x * → > [ st ( s ) μ ( ds ) x * is continuous in the X topology of X * and therefore ( V.3.9 ) is generated by some element x , e X ...
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 converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ