## Linear Operators, Part 1 |

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Page 245

Every linear operator on a

PROOF . Let { b1 , . . . , bn } be a Hamel basis for the

linear space X so that every x in X has a unique representation in the form x =

qbit ...

Every linear operator on a

**finite**dimensional normed linear space is continuous .PROOF . Let { b1 , . . . , bn } be a Hamel basis for the

**finite**dimensional normedlinear space X so that every x in X has a unique representation in the form x =

qbit ...

Page 246

Then the dimension of X * * is

X * * ( II . 3 . 19 ) , the dimension of X is

proof , X and X * have the same dimension . Q . E . D . In the preceding lemma it

is ...

Then the dimension of X * * is

**finite**, and , since X is equivalent to a subspace ofX * * ( II . 3 . 19 ) , the dimension of X is

**finite**. Hence , from the first part of thisproof , X and X * have the same dimension . Q . E . D . In the preceding lemma it

is ...

Page 290

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

of measurable sets of

En ) = L ( En , E ( En ) , y ) , we obtain a sequence { n } of functions in L . such that

...

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

**finite**, and let En be an increasing sequenceof measurable sets of

**finite**measure whose union is S . Using the theorem for L (En ) = L ( En , E ( En ) , y ) , we obtain a sequence { n } of functions in L . such that

...

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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