## Linear Operators: General theory |

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2 Lemma, (a) In a topological group G, any algebraic combination of any number

of variables xv . . ., xn is continuous as a map of Gx. . . xG into G. (b) In a

2 Lemma, (a) In a topological group G, any algebraic combination of any number

of variables xv . . ., xn is continuous as a map of Gx. . . xG into G. (b) In a

**linear****topological space**X, all linear combinations of any number of scalars *j, . . ., a^.Page 413

non-zero linear functional on the complex space X, which separates the sets M

and N. Q.E.D. 2.

1 are applied to

non-zero linear functional on the complex space X, which separates the sets M

and N. Q.E.D. 2.

**Linear Topological Spaces**In this section, the results of Section1 are applied to

**linear topological spaces**. Statements 1—6 of this section are ...Page 419

Let X be a

e) = fo||/(p)-/(?)| < e, J e A}, where p e X, A is a finite subset of T, and e > 0.

Let X be a

**linear**vector**space**, and let r be a total subspace of X+. Then the r**topology**of X is the**topology**obtained by taking as base all sets of the form N(p;A,e) = fo||/(p)-/(?)| < e, J e A}, where p e X, A is a finite subset of T, and e > 0.

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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