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Page 263
Since G is an arbitrary open set containing Fi - G , we have M ( F1 ) S2 ( G ) + M (
F1 - 6 ) . If F is a closed set it follows from this inequality , by allowing G , to range
over all open sets containing FF , that Mi ( Fi ) = Mz ( FF ; ) + Mz ( F1 - F ) .
Since G is an arbitrary open set containing Fi - G , we have M ( F1 ) S2 ( G ) + M (
F1 - 6 ) . If F is a closed set it follows from this inequality , by allowing G , to range
over all open sets containing FF , that Mi ( Fi ) = Mz ( FF ; ) + Mz ( F1 - F ) .
Page 269
Since G is an arbitrary open set containing Fi - G , we have M ( F1 ) S2 ( G ) + M (
F1 - G ) . If F is a closed set it follows from this inequality , by allowing G , to range
over all open sets containing FF1 , that M ( Fi ) 5M ( FF1 ) + 7 ( F1 - F ) . If E is an ...
Since G is an arbitrary open set containing Fi - G , we have M ( F1 ) S2 ( G ) + M (
F1 - G ) . If F is a closed set it follows from this inequality , by allowing G , to range
over all open sets containing FF1 , that M ( Fi ) 5M ( FF1 ) + 7 ( F1 - F ) . If E is an ...
Page 476
N ( T ; A , ε ) = { R | R € B ( X , Y ) , | ( T R ) xl < , X € A } where A is an arbitrary
finite subset of X , and ε > 0 is arbitrary . Thus , in the strong topology , a
generalized sequence { T , } converges to T if and only if { Tex } converges to Tx
for every x in ...
N ( T ; A , ε ) = { R | R € B ( X , Y ) , | ( T R ) xl < , X € A } where A is an arbitrary
finite subset of X , and ε > 0 is arbitrary . Thus , in the strong topology , a
generalized sequence { T , } converges to T if and only if { Tex } converges to Tx
for every x in ...
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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