## 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 3E

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 3E

is a

**normed linear space**, or a normed space, if to each x € 36 corresponds a real...

Page 65

For every x ^ 0 in a

= \x\. Proof. Apply Lemma 12 with 2J = 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 2J = 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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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions 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 Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact