## 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

Q.E.D. 2 THEOREM . ( Alaoglu ) The

...

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 -

with every positive multiple of the

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 intersectionwith 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

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 Yand ...

### 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