Linear Operators: General theory |
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Page 253
... Lemma 4 , M = A1 → ( M → A1 ) , and so M = A1 . The desired conclusion now follows from Theorem 10. Q.E.D. 13 THEOREM . For an orthonormal set ACH the following state- ments are equivalent : ( i ) the set A is complete ; ( ii ) the ...
... Lemma 4 , M = A1 → ( M → A1 ) , and so M = A1 . The desired conclusion now follows from Theorem 10. Q.E.D. 13 THEOREM . For an orthonormal set ACH the following state- ments are equivalent : ( i ) the set A is complete ; ( ii ) the ...
Page 422
... Theorem 2.10 , there exists a non- zero linear l - continuous functional g , and a real constant c , such that Rg ( ) c . By Lemma 1.11 , g ( $ , ) = 0 ; i.e. , f ( x ) = 0 implies g ( x ) = 0 . It follows from Lemma 10 that g af for ...
... Theorem 2.10 , there exists a non- zero linear l - continuous functional g , and a real constant c , such that Rg ( ) c . By Lemma 1.11 , g ( $ , ) = 0 ; i.e. , f ( x ) = 0 implies g ( x ) = 0 . It follows from Lemma 10 that g af for ...
Page 485
... follows from Lemma 7 and Lemma I.5.7 that T * S * is compact in the X ** topology of X * . Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the ...
... follows from Lemma 7 and Lemma I.5.7 that T * S * is compact in the X ** topology of X * . Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the ...
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