## 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 normed space, if to each x e X. corresponds a real...

Page 65

For every x 0 in a

= \x\. Proof. Apply Lemma 12 with 3) = 0. The x* required in the present corollary

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

...

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 3) = 0. The x* required in the present corollary

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

...

Page 91

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

Further, a

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

seen ...

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

Further, a

**normed linear space**is a B-spaee provided it is complete under someequivalent metric. Sec 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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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero