Linear Operators: General theory |
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Page 487
Operators with Closed Range It was observed in Lemma 2 . 8 that the closure of
the range of an operator U € B ( X , Y ) consists of those vectors y such that y * UX
= 0 implies y * y = 0 . Or , in other words , UX = { y \ U * y * = 0 implies y * y = 0 } .
Operators with Closed Range It was observed in Lemma 2 . 8 that the closure of
the range of an operator U € B ( X , Y ) consists of those vectors y such that y * UX
= 0 implies y * y = 0 . Or , in other words , UX = { y \ U * y * = 0 implies y * y = 0 } .
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
...
Page 513
( ii ) The range of U is closed if there exists a constant K such that for any y in the
range there exists a solution of y = Tx such that Kyl . ( iii ) U is one - to - one if the
range of U * is dense in X * . ( iv ) U * is one - to - one if and only if the range of U
...
( ii ) The range of U is closed if there exists a constant K such that for any y in the
range there exists a solution of y = Tx such that Kyl . ( iii ) U is one - to - one if the
range of U * is dense in X * . ( iv ) U * is one - to - one if and only if the range of U
...
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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 Volume 1 of Linear Operators, Jacob T. Schwartz Volume 7 of Pure and applied mathematics |
| Author | Nelson Dunford |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 16 Mar 2011 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |