## Linear Operators: General theory |

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Page 59

A linear space X is a

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

(i) |0| = 0; |*| > 0, x ^ 0; (ii) \x+y\ ^ |*| + |y|, x,yel; (iii) \<xx\ = |a||a:j, a € 0, xcdi.

A linear space X is a

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

(i) |0| = 0; |*| > 0, x ^ 0; (ii) \x+y\ ^ |*| + |y|, x,yel; (iii) \<xx\ = |a||a:j, a € 0, xcdi.

Page 65

For every x ^ 0 in a

= \x\. Proof. Apply Lemma 12 with ?) = 0. The x* required in the present corollary

may then be defined as \x\ times the x* whose existence is established in ...

For every x ^ 0 in a

**normed linear space**X, there is an x* e X* with \x*\ = 1 and x*x= \x\. Proof. Apply Lemma 12 with ?) = 0. The x* required in the present corollary

may then be defined as \x\ times the x* whose existence is established in ...

Page 245

Every linear operator on a finite dimensional

Proof. Let {bv . . ., bn} be a Hamel basis for the finite dimensional

Every linear operator on a finite dimensional

**normed linear space**is continuous.Proof. Let {bv . . ., 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 = a^-f.### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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