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 so that every x in X has a unique represen- a11 + ... + amon . It was shown in the ...
... finite dimensional normed linear space is continuous . PROOF . Let { b1 , . . . , bn } be a Hamel basis for the finite dimension- al normed linear space so that every x in X has a unique represen- a11 + ... + amon . It was shown in the ...
Page 246
... finite . Then the dimension of X ** is finite , and , since X is equivalent to a subspace of X ** ( II.3.19 ) , the dimension of X is finite . Hence , from the first part of this proof , X and X * have the same dimension . Q.E.D. In the ...
... finite . Then the dimension of X ** is finite , and , since X is equivalent to a subspace of X ** ( II.3.19 ) , the dimension of X is finite . Hence , from the first part of this proof , X and X * have the same dimension . Q.E.D. In the ...
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 ) ( En ) , μ ) , we obtain a sequence { g } of functions in L almost all s in En ∞ and = L1 ( En , such ...
... finite , and let E , be an increasing sequence of measurable sets of finite measure whose union is S. Using the theorem for L1 ( En ) ( En ) , μ ) , we obtain a sequence { g } of functions in L almost all s in En ∞ and = L1 ( En , such ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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