Linear Operators: General theory |
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Page 7
... shown that it is well - ordered and hence belongs to & . The statement a ' x for x € UE is clear . If x ≤'y and y ≤'z , then x , y € E。€ 80 , y , z € E1 € 80 , x Soy , and y ≤1 z . Since & is totally ordered it may be supposed that ...
... shown that it is well - ordered and hence belongs to & . The statement a ' x for x € UE is clear . If x ≤'y and y ≤'z , then x , y € E。€ 80 , y , z € E1 € 80 , x Soy , and y ≤1 z . Since & is totally ordered it may be supposed that ...
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 x 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 x be the smallest element of ca and let y be any ...
Page 553
... shown that in Hilbert space C is a maximal two - sided ideal . We have seen in Theorems 7.4 and 8.12 that in the spaces C and L1 , FCCC WCP . Grothendieck [ 4 ; p . 153 ] proved that in C , the ideals W and coin- cide . This is not the ...
... shown that in Hilbert space C is a maximal two - sided ideal . We have seen in Theorems 7.4 and 8.12 that in the spaces C and L1 , FCCC WCP . Grothendieck [ 4 ; p . 153 ] proved that in C , the ideals W and coin- cide . This is not the ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite 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 theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ