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Page 54
A linear mapping of one F - space into another is continuous if and only if it maps
bounded sets into bounded sets . PROOF . Let X , y be F - spaces , let T : X + y be
linear and continuous , and let B C # be bounded . For every neighborhood V ...
A linear mapping of one F - space into another is continuous if and only if it maps
bounded sets into bounded sets . PROOF . Let X , y be F - spaces , let T : X + y be
linear and continuous , and let B C # be bounded . For every neighborhood V ...
Page 97
il and to be bounded if u is bounded . The total variation v ( u ) of an additive set
function u is important because it dominates u in the sense that v ( u , E ) > lu ( E )
for E € E ; the reader should test his comprehension of Definition 4 below by ...
il and to be bounded if u is bounded . The total variation v ( u ) of an additive set
function u is important because it dominates u in the sense that v ( u , E ) > lu ( E )
for E € E ; the reader should test his comprehension of Definition 4 below by ...
Page 345
if and only if it is bounded and ( s ) converges ( to f ( s ) ) for each s in [ a , b ] and
each j = 0 , 1 , . . . , p . ( c ) CP is not weakly complete , and not reflexive . ( d ) A
subset A CCP is conditionally compact if and only if it is bounded and for each ...
if and only if it is bounded and ( s ) converges ( to f ( s ) ) for each s in [ a , b ] and
each j = 0 , 1 , . . . , p . ( c ) CP is not weakly complete , and not reflexive . ( d ) A
subset A CCP is conditionally compact if and only if it is bounded and for each ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
31 other sections not shown
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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