Linear Operators: General theory |
From inside the book
Results 1-3 of 77
Page 424
Since each projection is a continuous map , each o the sets A ( x , y ) and B ( « , x
) is closed . Hence tK = n x , ve ¥ A ( x , y ) nnaeyrex B ( a , x ) is also closed .
Q.E.D. 2 THEOREM . ( Alaoglu ) The closed unit sphere in the conjugate space X
...
Since each projection is a continuous map , each o the sets A ( x , y ) and B ( « , x
) is closed . Hence tK = n x , ve ¥ A ( x , y ) nnaeyrex B ( a , x ) is also closed .
Q.E.D. 2 THEOREM . ( Alaoglu ) The closed unit sphere in the conjugate space X
...
Page 429
( Krein - Šmulian ) A convex set in X * is X - closed if and only if its intersection
with every positive multiple of the closed unit sphere of X * is X - closed . Proof .
This follows from the preceding theorem and Corollary 2.14 . Q.E.D.
COROLLARY .
( Krein - Šmulian ) A convex set in X * is X - closed if and only if its intersection
with every positive multiple of the closed unit sphere of X * is X - closed . Proof .
This follows from the preceding theorem and Corollary 2.14 . Q.E.D.
COROLLARY .
Page 488
It follows from the definition of U * that every element in its range satisfies the
stated condition . Q.E.D. 3 LEMMA . If the adjoint of an operator U in B ( X , Y ) is
one - to - one and has a closed range , then UX = Y. Ux = PROOF . Let 0 #ye Y
and ...
It follows from the definition of U * that every element in its range satisfies the
stated condition . Q.E.D. 3 LEMMA . If the adjoint of an operator U in B ( X , Y ) is
one - to - one and has a closed range , then UX = Y. Ux = PROOF . Let 0 #ye Y
and ...
What people are saying - Write a review
We haven't found any reviews in the usual places.
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
80 other sections not shown
Common terms and phrases
algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero
References to this book
Bibliographic information
| Title | Linear Operators: General theory Part 1 of Linear Operators, Jacob T. Schwartz Volume 1; Volume 7 of Pure and applied mathematics : a series of texts and monographs Volume 7 of Pure and applied mathematics Interscience Press Pure and applied mathematics, ISSN 0079-8185 Wiley classics library |
| Authors | Nelson Dunford, Jacob T. Schwartz, William G. Bade |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of Michigan |
| Digitized | 20 Nov 2007 |
| ISBN | 0470226056, 9780470226056 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |