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 X so that every x in X has a unique represen- tation in the form x a11 + ... + anbn ...
... finite dimensional normed linear space is continuous . PROOF . Let { b1 , · .... b } 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 ...
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 ( 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 ( En ) , μ ) , we obtain a sequence { g } of functions in L such that gn∞ ≤ x * , gn ( 8 ) ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ