## Linear Operators: General theory |

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Page 23

If f : A -> Y is uniformly continuous on the dense subset A of X, then f has a

continuous extension g : X -> Y. This

X. Proof. If x e X, there is a sequence of 1.6.16 23 METRIC SPACES.

If f : A -> Y is uniformly continuous on the dense subset A of X, then f has a

**unique**continuous extension g : X -> Y. This

**unique**extension is uniformly continuous onX. Proof. If x e X, there is a sequence of 1.6.16 23 METRIC SPACES.

Page 202

We will first show that fi is

with the stated value on elementary sets. For each 71 let k„ be set functions on Z„

defined by the formulas f*M = n(E„ x S„,), k„(E„) = k(E„ x Snl), En e Z„. Then (zc7[ ...

We will first show that fi is

**unique**. Let k be another additive set function on E1with the stated value on elementary sets. For each 71 let k„ be set functions on Z„

defined by the formulas f*M = n(E„ x S„,), k„(E„) = k(E„ x Snl), En e Z„. Then (zc7[ ...

Page 564

+oo, and a solution is a complex valued (column) vector y(t) e 22", differentiable

and satisfying the differential system for each tel. The theory of differential

equations assures us that for each t0 e I and each y0 e En there exists a

solution ...

+oo, and a solution is a complex valued (column) vector y(t) e 22", differentiable

and satisfying the differential system for each tel. The theory of differential

equations assures us that for each t0 e I and each y0 e En there exists a

**unique**solution ...

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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