Linear Operators: General theory |
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Page 422
... Theorem 2.10 , there exists a non- zero linear T - 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 T - 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 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 ...
... 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 ...
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
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ