## Linear Operators, Part 1 |

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Page 424

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

) is

. D . 2 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 = Ox , vex A ( x , y ) anae , rex B ( a , 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 - Šmulian ) A convex set in X * is X -

only if its intersection with every positive multiple of the

is X -

Q . E . D . 7 THEOREM . ( Krein - Šmulian ) A convex set in X * is X -

**closed**if andonly 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

If the adjoint of an operator U in B ( X , Y ) is one - to - one and has a

range , then UX = Y . PROOF . Let 0 # ye Y and define [ = { y * y * e Y * , y * y = 0 }

. Then I ' is Y -

and ...

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 # ye Y and define [ = { y * y * e Y * , y * y = 0 }

. Then I ' is Y -

**closed**in Y * . Suppose , for the moment , that U * l ' is X -**closed**and ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

Akad 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 isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric 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