## 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 ...

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 ...

Page 245

An n-dimensional B-space is equivalent to E". 4 Corollary. Every linear operator

on a finite dimensional

be a Hamel basis for the finite dimensional

...

An n-dimensional B-space is equivalent to E". 4 Corollary. Every linear operator

on a finite dimensional

**normed linear space**is continuous. Proof. Let {bv . . ., bn}be a Hamel basis for the finite dimensional

**normed linear space**X so that every x...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero