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Page 60
Statement ( iv ) clearly implies the continuity of T at 0 ; so ( iv ) implies ( ii ) . This ( i ) , ( ii ) , and ( iv ) are equivalent . If M = sup Tx | is finite , then for an arbitrary x + 0 , = 0 1351 X Tx } = ( x T ( ) SMX !
Statement ( iv ) clearly implies the continuity of T at 0 ; so ( iv ) implies ( ii ) . This ( i ) , ( ii ) , and ( iv ) are equivalent . If M = sup Tx | is finite , then for an arbitrary x + 0 , = 0 1351 X Tx } = ( x T ( ) SMX !
Page 280
That ( 1 ) implies ( 2 ) can be proved in a manner similar to that used in Theorem 14 to show that condition ( 3 ) of that theorem implies ( 4 ) . From Corollary 19 it follows that S may be embedded as a dense subset of a compact ...
That ( 1 ) implies ( 2 ) can be proved in a manner similar to that used in Theorem 14 to show that condition ( 3 ) of that theorem implies ( 4 ) . From Corollary 19 it follows that S may be embedded as a dense subset of a compact ...
Page 394
A real B - space is said to be an abstract L - space if it is a vector lattice in which ( 1 ) x 20 and y 20 imply ( x + y ... an equivalent norm which satisfies ( 1 ) and also the condition ( m ) X A Y = 0 O implies ( x + y ) = | - yl .
A real B - space is said to be an abstract L - space if it is a vector lattice in which ( 1 ) x 20 and y 20 imply ( x + y ... an equivalent norm which satisfies ( 1 ) and also the condition ( m ) X A Y = 0 O implies ( x + y ) = | - yl .
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
quences | 26 |
Copyright | |
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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc Proof properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero