## Linear Operators, Part 1 |

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Page 59

A linear space 3 is a

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

i) |0| = 0; a > 0, a # 0; (ii) a +y| = |x|+|y|. a, ye &; (iii) loor = |x||a|, ... e D, a e 3.

A linear space 3 is a

**normed linear space**, or a normed space, if to each a e 3corresponds a real number a called the norm of a which satisfies the conditions: (

i) |0| = 0; a > 0, a # 0; (ii) a +y| = |x|+|y|. a, ye &; (iii) loor = |x||a|, ... e D, a e 3.

Page 65

For every a # 0 in a

wow = |a|. PRoof. Apply Lemma 12 with 9) = 0. The ao required in the present

corollary may then be defined as a times the a” whose existence is established in

...

For every a # 0 in a

**normed linear space**3, there is an iro e 3" with wo – 1 andwow = |a|. PRoof. Apply Lemma 12 with 9) = 0. The ao required in the present

corollary may then be defined as a times the a” whose existence is established in

...

Page 91

Thus every complete linear metric space can be metrized to be an F-space.

Further, a

equivalent metric. See also van Dantzig [1], [2]. Norms in linear spaces. We have

seen ...

Thus every complete linear metric space can be metrized to be an F-space.

Further, a

**normed linear space**is a B-space provided it is complete under someequivalent metric. See also van Dantzig [1], [2]. Norms in linear spaces. We have

seen ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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