## Linear Operators: General theory |

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Page 145

A sequence of functions {/„} defined on S with values in 3: converges u-

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

converges

function f ...

A sequence of functions {/„} defined on S with values in 3: converges u-

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

converges

**uniformly**on S — E. The sequence {/„} converges /u-**uniformly**to thefunction f ...

Page 314

A subset K of ba(S, 27) is weakly sequentially compact if and only if there exists a

non-negative /< in ba(S, 27) such that lim k(E) = 0

K Q ba(S, 27) be weakly sequentially compact and let V = UT be the isometric ...

A subset K of ba(S, 27) is weakly sequentially compact if and only if there exists a

non-negative /< in ba(S, 27) such that lim k(E) = 0

**uniformly**for A e K. Proof. LetK Q ba(S, 27) be weakly sequentially compact and let V = UT be the isometric ...

Page 360

which vanishes in a neighborhood of a point p the sequence (5„/)(x) converges to

zero

then the convergence of S„f for a given c.o.n. system is localized if and only if ...

which vanishes in a neighborhood of a point p the sequence (5„/)(x) converges to

zero

**uniformly**for x in some neighborhood of p. 21 Show that if | J"J En(x, z)dz\ M,then the convergence of S„f for a given c.o.n. system is localized if and only if ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### 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 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