## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 81

Page 424

Since each projection is a continuous map , each o the sets A ( x , y ) and B ( a , x

) is

THEOREM . ( Alaoglu ) The

Since each projection is a continuous map , each o the sets A ( x , y ) and B ( a , x

) is

**closed**. Hence tK = n . , vex A ( x , y ) nnacoret B ( a , x ) is also**closed**. Q.E.D.THEOREM . ( Alaoglu ) The

**closed**unit sphere in the conjugate space X * of ...Page 489

since the range of U * is

restriction of y * to 3 , then w * U * r * . Hence , the range of U * is also

follows from the previous lemma that U.X UX 3 . Hence , U has a

Q.E.D. 5 ...

since the range of U * is

**closed**, ** = U * y * for some y * € Y * . If z * is therestriction of y * to 3 , then w * U * r * . Hence , the range of U * is also

**closed**. Itfollows from the previous lemma that U.X UX 3 . Hence , U has a

**closed**range .Q.E.D. 5 ...

Page 513

( ii ) The range of U is

range there exists a solution of y Tx such that \ x \ S 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 ...

( ii ) The range of U is

**closed**if there exists a constant K such that for any y in therange there exists a solution of y Tx such that \ x \ S 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 ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero