## 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 G X . . . xG into G. (b) In a linear

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

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

**topological space**X, all linear combinations of any number of scalars ar . . ., 0^, ...Page 413

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

and N. Q.E.D. 2. Linear

1 are applied to linear

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

2 Definition. Let X be a linear vector

Then the r

form N(p; A,e) = {?||/(p)-/(g)| < e, / , A}, where p e X, A is a finite subset of 71, and e

> ...

2 Definition. Let X be a linear vector

**space**, and let f be a total subspaee of X+.Then the r

**topology**of X is the**topology**obtained by taking as base all sets of theform N(p; A,e) = {?||/(p)-/(g)| < e, / , A}, where p e X, A is a finite subset of 71, and e

> ...

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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