Linear Operators: General theory |
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Page 245
... finite dimensional normed linear space is continuous . PROOF . Let { b1 , bn } be a Hamel basis for the finite dimension- al normed linear space X so that every x in X has a unique represen- tation in the form x = a11 + ... + anbn . It ...
... finite dimensional normed linear space is continuous . PROOF . Let { b1 , bn } be a Hamel basis for the finite dimension- al normed linear space X so that every x in X has a unique represen- tation in the form x = a11 + ... + anbn . It ...
Page 290
... finite , and let E , be an increasing sequence of measurable sets of finite measure whose union is S. Using the theorem for L1 ( En ) L1 ( En , Σ ( En ) , μ ) , we obtain a sequence { g } of functions in L such that gn * , gn ( 8 ) gn + ...
... finite , and let E , be an increasing sequence of measurable sets of finite measure whose union is S. Using the theorem for L1 ( En ) L1 ( En , Σ ( En ) , μ ) , we obtain a sequence { g } of functions in L such that gn * , gn ( 8 ) gn + ...
Page 849
... finite , III.4.3 ( 126 ) Lebesgue extension of , III.5.18 ( 143 ) as a metric space , III.7.1 ( 158 ) , III.9.6 ( 169 ) positive , III.4.3 ( 126 ) product , of finite number of finite measure spaces , III.11.3 ( 186 ) of finite number of o ...
... finite , III.4.3 ( 126 ) Lebesgue extension of , III.5.18 ( 143 ) as a metric space , III.7.1 ( 158 ) , III.9.6 ( 169 ) positive , III.4.3 ( 126 ) product , of finite number of finite measure spaces , III.11.3 ( 186 ) of finite number of o ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ