## Linear Operators: General theory |

### From inside the book

Results 1-3 of 90

Page 14

The topology in the set of scalars is always assumed to be that determined by the

base whose elements are neighborhoods of the form {/3||/3— <x| < e}. ... Let f, g

be continuous scalar functions on a

The topology in the set of scalars is always assumed to be that determined by the

base whose elements are neighborhoods of the form {/3||/3— <x| < e}. ... Let f, g

be continuous scalar functions on a

**topological space**X, and let v be a scalar.Page 50

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

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

„ ...

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

of variables xv . . n 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 scalar s ocj, . . ., <x„„ ...

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 ...### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

78 other sections not shown

### Other editions - View all

### Common terms and phrases

additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations disjoint Doklady Akad Duke Math element ergodic theorem exists finite dimensional function defined Hausdorff space Hence Hilbert space homeomorphism ibid inequality integral Lebesgue Lebesgue measure Lemma linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc Proof properties proved real numbers reflexive Riesz Russian Sbornik N. S. scalar self-adjoint semi-group sequentially compact Show simple functions spectral Studia Math subset subspace Suppose theory topological space Trans uniformly Univ valued function Vber vector space weak topology weakly compact zero