Linear Operators: General theory |
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Page 54
... 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 of the zero in Y , there is a neighborhood U of zero in X such that T ( U ) ...
... 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 of the zero in Y , there is a neighborhood U of zero in X such that T ( U ) ...
Page 345
... bounded and the set { f ( P ) } , fe A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in A ( D ) is a weak Cauchy sequence ( a sequence converging weakly to f in A ( D ) ) if and only if ...
... bounded and the set { f ( P ) } , fe A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in A ( D ) is a weak Cauchy sequence ( a sequence converging weakly to f in A ( D ) ) if and only if ...
Page 436
... bounded ( II.1.7 ) set , then X must be finite dimensional . 6 Let X be a B - space , and X , a subspace of X. Show ... bounded if and only if f ( A ) is a bounded set of scalars for each fel . 8 Let X be a B - space . Show that a subset ...
... bounded ( II.1.7 ) set , then X must be finite dimensional . 6 Let X be a B - space , and X , a subspace of X. Show ... bounded if and only if f ( A ) is a bounded set of scalars for each fel . 8 Let X be a B - space . Show that a subset ...
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