Linear Operators: General theory |
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Page 51
If a homomorphism of one topological group into another is continuous anywhere
, it is continuous . Proof . Let the homomorphism f : G → H be continuous at x ,
and let y eG . If V is a neighborhood of f ( y ) , then , by Lemma 2 ( c ) , V ] ( y - 1x )
...
If a homomorphism of one topological group into another is continuous anywhere
, it is continuous . Proof . Let the homomorphism f : G → H be continuous at x ,
and let y eG . If V is a neighborhood of f ( y ) , then , by Lemma 2 ( c ) , V ] ( y - 1x )
...
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 572
Conversely , let f ( T ) = 0 ; then , by Theorem 11 , flo ( T ) ) = 0 . Let | be analytic
on a neighborhood U of o ( T ) . For each Q E O ( T ) , there is an ela ) > 0 such
that the sphere S ( Q , Ela ) ) CU . Since o ( T ) is compact , a finite set of spheres
S ...
Conversely , let f ( T ) = 0 ; then , by Theorem 11 , flo ( T ) ) = 0 . Let | be analytic
on a neighborhood U of o ( T ) . For each Q E O ( T ) , there is an ela ) > 0 such
that the sphere S ( Q , Ela ) ) CU . Since o ( T ) is compact , a finite set of spheres
S ...
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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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Common terms and phrases
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
Bibliographic information
| Title | Linear Operators: General theory Part 1 of Linear Operators, Jacob T. Schwartz Volume 7 of Pure and applied mathematics Interscience Press Pure and applied mathematics Wiley classics library |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 16 Mar 2011 |
| ISBN | 0470226056, 9780470226056 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |