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 AC the following state- ments are equivalent : ( i ) the set A is complete ; ( ii ) the set ...
... 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 AC the following state- ments are equivalent : ( i ) the set A is complete ; ( ii ) the set ...
Page 435
... Theorem 3.13 , it follows from Theorem 1 that it suffices to show that co ( 4 ) is weakly sequentially compact . Let { P } be a sequence of points in co ( A ) ; then p , 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 ( 4 ) is weakly sequentially compact . Let { P } be a sequence of points in co ( A ) ; then p , is a convex combina tion of a finite set B of points of A. Let B ...
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 X ...
... 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 X ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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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 Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ