## Linear Operators: General theory |

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Page 24

Since {a„} is a Cauchy sequence, and / is

„)} is also a Cauchy sequence. Since Y is complete, there is a point g(x) e Y with f

(a„) -> g(x). To verify that g(x) depends only upon x, and not upon the particular ...

Since {a„} is a Cauchy sequence, and / is

**uniformly**continuous, the sequence {/(a„)} is also a Cauchy sequence. Since Y is complete, there is a point g(x) e Y with f

(a„) -> g(x). To verify that g(x) depends only upon x, and not upon the particular ...

Page 145

A sequence of functions {/„} defined on S with values in i converges u-

for each e > 0 there is a set E e Z such that v(fi, E) < e and such that {/„} converges

A sequence of functions {/„} defined on S with values in i converges u-

**uniformly**iffor each e > 0 there is a set E e Z such that v(fi, E) < e and such that {/„} converges

**uniformly**on S — E. The sequence {/„} converges fi-**uniformly**to tlie function f if ...Page 314

Q.E.D. 12 Theorem. A subset K of ba(S, E) is weakly sequentially compact if arid

only if there exists a non-negative u in ba(S, E) such that lim X(E) = 0

X e K. Proof. Let K Q ba(S, E) be weakly sequentially compact and let V = UT be ...

Q.E.D. 12 Theorem. A subset K of ba(S, E) is weakly sequentially compact if arid

only if there exists a non-negative u in ba(S, E) such that lim X(E) = 0

**uniformly**forX e K. Proof. Let K Q ba(S, E) be weakly sequentially compact and let V = UT be ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero