Linear Operators, Part 1 |
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Page 412
10 LEMMA . The linear functional f separates the subsets M and N of X if and only
if it separates the subsets M - N and { 0 } of X . The proof is elementary , and is left
to the reader . In dealing with subspaces , it is often convenient to make use of ...
10 LEMMA . The linear functional f separates the subsets M and N of X if and only
if it separates the subsets M - N and { 0 } of X . The proof is elementary , and is left
to the reader . In dealing with subspaces , it is often convenient to make use of ...
Page 418
If p and q are distinct points of a locally convex linear topological space X , there
is a continuous linear functional ... Tz ) have the same continuous linear
functionals , then a convex set is closed in ( X , 17 ) if and only if it is closed in ( X ,
tz ) .
If p and q are distinct points of a locally convex linear topological space X , there
is a continuous linear functional ... Tz ) have the same continuous linear
functionals , then a convex set is closed in ( X , 17 ) if and only if it is closed in ( X ,
tz ) .
Page 421
Let X be a linear space , and let l be a total subspace of X * . Then the linear
functionals on X which are continuous in the l ' topology are precisely the
functionals in . . The proof of Theorem 9 will be based on the following lemma .
10 LEMMA .
Let X be a linear space , and let l be a total subspace of X * . Then the linear
functionals on X which are continuous in the l ' topology are precisely the
functionals in . . The proof of Theorem 9 will be based on the following lemma .
10 LEMMA .
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
25 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 obtained 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, Part 1 Linear Operators, Jacob T. Schwartz Volume 7 of Pure and applied mathematics, ISSN 0079-8185 Volume 7 of Pure and applied mathematics. A series of texts and monographs |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 16 Mar 2011 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |