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 493
... mapping which sends a * into u ( · ) x * , then it follows that for each fixed ƒ € C ( S ) the mapping X * → → [ ç † ( s ) μ ( ds ) x * is continuous in the X topology of X * and therefore ( V.3.9 ) is generated by some element x , e ...
... mapping which sends a * into u ( · ) x * , then it follows that for each fixed ƒ € C ( S ) the mapping X * → → [ ç † ( s ) μ ( ds ) x * is continuous in the X topology of X * and therefore ( V.3.9 ) is generated by some element x , e ...
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 topology ...
... 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 topology ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ