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 ACS 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 ACS the following state- ments are equivalent : ( i ) the set A is complete ; ( ii ) the ...
Page 422
... Lemma 1.11 , g ( , ) = 0 ; i.e. , f ( x ) ≤ 0 ; i.e. , f ( x ) = 0 implies g ( x ) = 0 . af for some non - zero scalar z . By It follows from Lemma 10 that g Theorem 9 , g is in I ; thus f is in I. Q.E.D. 12 COROLLARY . Let X be a ...
... Lemma 1.11 , g ( , ) = 0 ; i.e. , f ( x ) ≤ 0 ; i.e. , f ( x ) = 0 implies g ( x ) = 0 . af for some non - zero scalar z . By It follows from Lemma 10 that g Theorem 9 , g is in I ; thus f is in I. Q.E.D. 12 COROLLARY . Let X be a ...
Page 435
... Theorem 3.13 , it follows from Theorem 1 that it suffices to show that co ( A ) is weakly sequentially compact . Let { P } be a sequence of points in co ( A ) ; then p1 is a convex combina tion of a finite set B , of points of A. Let B ...
... Theorem 3.13 , it follows from Theorem 1 that it suffices to show that co ( A ) is weakly sequentially compact . Let { P } be a sequence of points in co ( A ) ; then p1 is a convex combina tion of a finite set B , of points of A. Let B ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ