## Linear Operators: General theory |

### From inside the book

Results 1-3 of 80

Page 245

Every linear operator on a

Every linear operator on a

**finite**dimensional normed linear space is continuous . PROOF . Let { bi , . bn } be a Hamel basis for the**finite**dimensional normed linear space X so that every x in X has a unique representation in the form x ...Page 246

Then the dimension of X ** is

Then the dimension of X ** is

**finite**, and , since X is equivalent to a subspace of X ** ( II.3.19 ) , the dimension of X is**finite**. Hence , from the first part of this proof , X and X * have the same dimension .Page 290

Now suppose that ( S , E , j ) is o -

Now suppose that ( S , E , j ) is o -

**finite**, and let En be an increasing sequence of measurable sets of**finite**measure whose union is S. Using the theorem for L ( En ) = L ( En , E ( En ) , u ) , we obtain a sequence { gn } of ...### 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

### Other editions - View all

### 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 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 u-measurable uniform uniformly unique unit valued vector weak weakly compact zero