## Linear Operators: General theory |

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Page 59

A linear space X is a

corresponds a real number la called the norm of x which satisfies the conditions :

( i ) 101 = 0 ; \ Q1 > 0 , x = 0 ; ( ii ) læ + y = x + yl , X , Y e X ; ( iii ) ox | = ollal , QE Ø

...

A linear space X is a

**normed linear space**, or a normed space , if to each x e Xcorresponds a real number la called the norm of x which satisfies the conditions :

( i ) 101 = 0 ; \ Q1 > 0 , x = 0 ; ( ii ) læ + y = x + yl , X , Y e X ; ( iii ) ox | = ollal , QE Ø

...

Page 65

Nelson Dunford, Jacob T. Schwartz. 14 COROLLARY . For every x + 0 in a

. Apply Lemma 12 with Y = 0 . The æ * required in the present corollary may then

be ...

Nelson Dunford, Jacob T. Schwartz. 14 COROLLARY . For every x + 0 in a

**normed linear space**X , there is an æ * X * with 12 * 1 = 1 and x * x = \ al . PROOF. Apply Lemma 12 with Y = 0 . The æ * required in the present corollary may then

be ...

Page 245

An n - dimensional B - space is equivalent to En . 4 COROLLARY . Every linear

operator on a finite dimensional

{ b , . . . , bn } be a Hamel basis for the finite dimensional

An n - dimensional B - space is equivalent to En . 4 COROLLARY . Every linear

operator on a finite dimensional

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

**normed linear space**X ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential 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 mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero