Linear Operators: General theory |
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Page 240
The space bs is the linear space of all sequences x = { en } of scalars for which
the norm ( x ) = sup / Žail is finite . 11 . The space ... SES A scalar function f on S
is E - measurable if f - 1 ( A ) ¢ { for every Borel set A in the range of f . It is clear
that ...
The space bs is the linear space of all sequences x = { en } of scalars for which
the norm ( x ) = sup / Žail is finite . 11 . The space ... SES A scalar function f on S
is E - measurable if f - 1 ( A ) ¢ { for every Borel set A in the range of f . It is clear
that ...
Page 256
To summarize , we state the following definition . 17 DEFINITION . For each i = 1 ,
. . . , n , let H ; be a Hilbert space with scalar products ( : , ) i . The direct sum of the
Hilbert spaces Hi , . . . , Hn is the linear space H = H , 0 . . . Hnin which a scalar ...
To summarize , we state the following definition . 17 DEFINITION . For each i = 1 ,
. . . , n , let H ; be a Hilbert space with scalar products ( : , ) i . The direct sum of the
Hilbert spaces Hi , . . . , Hn is the linear space H = H , 0 . . . Hnin which a scalar ...
Page 323
A scalar valued measurable function f is said to be integrable if there exists a
sequence { n } of simple functions such that ( i ) In ( s ) converges to f ( s ) 4 -
almost everywhere ; ( ii ) the sequence { Seln ( s ) u ( ds ) } converges in the norm
of X for ...
A scalar valued measurable function f is said to be integrable if there exists a
sequence { n } of simple functions such that ( i ) In ( s ) converges to f ( s ) 4 -
almost everywhere ; ( ii ) the sequence { Seln ( s ) u ( ds ) } converges in the norm
of X for ...
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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