Linear Operators: General theory |
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Page 245
... shown in the proof of tation in the form a Lemma 1 that a ,, i = 1 , . . . , n , depends continuously upon æ . Thus if U is a linear operator on X it is seen that Ux also continuous in x . Q.E.D. = ¤1Üb1 + . . . + ɑ „ Ub „ is 5 THEOREM ...
... shown in the proof of tation in the form a Lemma 1 that a ,, i = 1 , . . . , n , depends continuously upon æ . Thus if U is a linear operator on X it is seen that Ux also continuous in x . Q.E.D. = ¤1Üb1 + . . . + ɑ „ Ub „ is 5 THEOREM ...
Page 335
... shown that W satisfies the hypothesis of Zorn's lemma . To do this we let Wo be a totally ordered subset of W ( I.22 ) and let c CuW 。• Then , for some a e Wo , ca is not void . Let a be the smallest element of ca and let y be any ...
... shown that W satisfies the hypothesis of Zorn's lemma . To do this we let Wo be a totally ordered subset of W ( I.22 ) and let c CuW 。• Then , for some a e Wo , ca is not void . Let a be the smallest element of ca and let y be any ...
Page 479
... shown that TX . If ye and ye TX , there is ( II.3.13 ) a y * in * with y * 0 , y * T = T * y * 0. This contradicts the assumption that T * is one - to - one , and proves the lemma . Q.E.D. The final argument in the above proof also ...
... shown that TX . If ye and ye TX , there is ( II.3.13 ) a y * in * with y * 0 , y * T = T * y * 0. This contradicts the assumption that T * is one - to - one , and proves the lemma . Q.E.D. The final argument in the above proof also ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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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