## Linear Operators, Part 1 |

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

of variables a 1, ..., a, is continuous as a map of GX. . . × G into G. (b) In a

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

of variables a 1, ..., a, is continuous as a map of GX. . . × G into G. (b) In a

**linear****topological space**3., all linear combinations of any number of scalars x1, ..., xm, ...Page 413

non-zero linear functional on the complex space 3., which separates the sets M

and N. Q.E.D. 2.

1 are applied to

non-zero linear functional on the complex space 3., 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 3: be a

, e) = {q|f(p)-f(q) < e, fe A), where p e 3, A is a finite subset of T, and e > 0. The

terms ...

Let 3: be a

**linear**vector**space**, and let T be a total subspace of £". Then the T**topology**of 3 is the**topology**obtained by taking as base all sets of the form N(p; A, e) = {q|f(p)-f(q) < e, fe A), where p e 3, A is a finite subset of T, and e > 0. The

terms ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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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