## Linear Operators: General theory |

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Page 245

Every linear operator on a

Proof . Let { bı , . . . bn } be a Hamel basis for the

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

Every linear operator on a

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

**finite**dimensional normed linearspace X so that every x in X has a unique representation in the form x = Qzbat ...

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 849

3 ( 126 )

metric space , I11 . 7 . 1 ( 158 ) , III . 9 . 6 ( 169 ) positive , III . 4 . 3 ( 126 ) product ,

of

3 ( 126 )

**finite**, III . 4 . 3 ( 126 ) Lebesgue extension of , 111 . 5 . 18 ( 143 ) as ametric space , I11 . 7 . 1 ( 158 ) , III . 9 . 6 ( 169 ) positive , III . 4 . 3 ( 126 ) product ,

of

**finite**number of**finite**measure spaces , III . 11 . 3 ( 186 ) of**finite**number of o ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

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