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Page 20
Thus A , + X , and A is open , and contains a sphere S , = S ( P1 , & j ) with 0 < Ei
< 1 / 2 . By assumption , the set A2 does not contain the open set S ( P1 , & z / 2 ) ;
hence the non - void open set An S ( P1 , & 1 / 2 ) contains a sphere S , = S ( P2 ...
Thus A , + X , and A is open , and contains a sphere S , = S ( P1 , & j ) with 0 < Ei
< 1 / 2 . By assumption , the set A2 does not contain the open set S ( P1 , & z / 2 ) ;
hence the non - void open set An S ( P1 , & 1 / 2 ) contains a sphere S , = S ( P2 ...
Page 56
Nelson Dunford, Jacob T. Schwartz. of the image of any neighborhood G of the
element 0 in X contains a neighborhood of the element 0 in Y . Since a - b is a
continuous function of a and b , there is a neighborhood M of 0 such that M –
MCG .
Nelson Dunford, Jacob T. Schwartz. of the image of any neighborhood G of the
element 0 in X contains a neighborhood of the element 0 in Y . Since a - b is a
continuous function of a and b , there is a neighborhood M of 0 such that M –
MCG .
Page 309
The set A = S - UAE ; then contains no atoms having measure greater than ε . It
will now be shown that every measurable subset B of A contains a set F with 0 <
u ( F ) Sε . Suppose , on the contrary , that some such set B contains no set of with
...
The set A = S - UAE ; then contains no atoms having measure greater than ε . It
will now be shown that every measurable subset B of A contains a set F with 0 <
u ( F ) Sε . Suppose , on the contrary , that some such set B contains no set of with
...
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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