Linear Operators: General theory |
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Page 424
Since each projection is a continuous map , each o the sets A ( x , y ) and B ( a ,
æ ) is closed . Hence tK = nt , vex A ( x , y ) onde pcex B ( « , 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 ( a ,
æ ) is closed . Hence tK = nt , vex A ( x , y ) onde pcex B ( « , x ) is also closed . Q .
E . D . 2 THEOREM . ( Alaoglu ) The closed unit sphere in the conjugate space X
...
Page 429
Q . E . D . 7 THEOREM . ( Krein - Smulian ) 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 .
Q . E . D . 7 THEOREM . ( Krein - Smulian ) 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 .
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 . PROOF . Let 0 + y e 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 . PROOF . Let 0 + y e Y and
...
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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
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 |