Linear Operators: General theory |
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Page 245
... finite dimensional normed linear space is continuous . PROOF . Let { b1 , . . . , b , } be a Hamel basis for the finite dimension- al normed linear space so that every x in X has a unique represen- tation in the form x = a1b1 + ... + ...
... finite dimensional normed linear space is continuous . PROOF . Let { b1 , . . . , b , } be a Hamel basis for the finite dimension- al normed linear space so that every x in X has a unique represen- tation in the form x = a1b1 + ... + ...
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 ) L1 ( En , Σ ( En ) , μ ) , we obtain a sequence { g } of functions in L such that gn≤ x * , gn ( 8 ) ...
... 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≤ x * , gn ( 8 ) ...
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