Linear Operators: General theory |
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Page 421
It is obvious that y is a linear functional on the subspace T ( X ) of En . By II.3.11 , it can be extended to a linear functional 4. on E . By IV.3.7 , 41 has the form Wi [ Y1 , . . . , Yn ] ] = { diyi Hence n i = 1 g ( x ) = 2a ( ) ...
It is obvious that y is a linear functional on the subspace T ( X ) of En . By II.3.11 , it can be extended to a linear functional 4. on E . By IV.3.7 , 41 has the form Wi [ Y1 , . . . , Yn ] ] = { diyi Hence n i = 1 g ( x ) = 2a ( ) ...
Page 423
Therefore , T'is weakly continuous at the origin , and hence at every point . Conversely , suppose that T is weakly continuous , and y * € Y * . Then y * T is a linear functional on X which is X * -continuous . Hence , by Theorem 9 ...
Therefore , T'is weakly continuous at the origin , and hence at every point . Conversely , suppose that T is weakly continuous , and y * € Y * . Then y * T is a linear functional on X which is X * -continuous . Hence , by Theorem 9 ...
Page 485
Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the X * , Y *** topologies in X ** , Y ** , respectively . If S , S ** are the closed unit spheres in 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 * , Y *** topologies in X ** , Y ** , respectively . If S , S ** are the closed unit spheres in X ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
quences | 26 |
Copyright | |
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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero