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 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 . 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 ...
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 , nuous , and let ...
... 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 , nuous , and let ...
Page 513
... mapping T → T * of B ( S ) 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 ( S ) 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 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ