## Linear Operators: General theory |

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Page 59

scalar a and every vector x. An investigation into the consequences of this

identity is the chief purpose of the present section. 1 Definition. A linear space X

is a

...

scalar a and every vector x. An investigation into the consequences of this

identity is the chief purpose of the present section. 1 Definition. A linear space X

is a

**normed linear space**, or a nortned space, if to each x e X corresponds a real...

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 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

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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