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 . 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 . - > 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 . - > 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
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ