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Page 421
7 , 4 , has the form Vi lyp - - „ Yn ] = { x : Si Hence 8 ( x ) = { viti ( x ) , X e X . i = 1 Q
. E . D . PROOF OF THEOREM 9 . Every functional in I ' is I - continuous , by
Lemma 8 . Conversely , let g + 0 be a linear functional on X which is I ' -
continuous .
7 , 4 , has the form Vi lyp - - „ Yn ] = { x : Si Hence 8 ( x ) = { viti ( x ) , X e X . i = 1 Q
. E . D . PROOF OF THEOREM 9 . Every functional in I ' is I - continuous , by
Lemma 8 . Conversely , let g + 0 be a linear functional on X which is I ' -
continuous .
Page 423
Hence \ y * ( Tx ) | < € , so that Tx e N ( 0 ; yt , . . . , Y , € ) . Therefore , Tis 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
...
Hence \ y * ( Tx ) | < € , so that Tx e N ( 0 ; yt , . . . , Y , € ) . Therefore , Tis 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
...
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 , X * * ,
respectively ...
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 , X * * ,
respectively ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero