## Linear Operators, Part 1 |

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Page 59

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

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

3 is a

real ...

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

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

3 is a

**normed linear space**, or a normed space, if to each a e 3 corresponds areal ...

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 | |

27 other sections not shown

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additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero