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Page 91
Thus every complete linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] .
Thus every complete linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] .
Page 239
The space Inc is the linear space of all ordered n - tuples [ cy , . .. , Cen ] of scalars Oy , ... , Olin with the norm lv sup laila 1 Sisn 4. The space l , is defined for 1 sp < oo as the linear space of all sequences x = { an } of ...
The space Inc is the linear space of all ordered n - tuples [ cy , . .. , Cen ] of scalars Oy , ... , Olin with the norm lv sup laila 1 Sisn 4. The space l , is defined for 1 sp < oo as the linear space of all sequences x = { an } of ...
Page 410
The intersection of an arbitrary family of convex subsets of the linear space X is convex. As examples of convex subsets of X, we note the subspaces of X, and the subsets of X consisting of one point. 3 Lemma.
The intersection of an arbitrary family of convex subsets of the linear space X is convex. As examples of convex subsets of X, we note the subspaces of X, and the subsets of X consisting of one point. 3 Lemma.
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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