## Linear Operators, Part 1 |

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Page 20

A

} in a metric space is a Cauchy

Cauchy

...

A

**sequence**{ an } is said to be convergent if an → a for some a . A**sequence**{ an} in a metric space is a Cauchy

**sequence**if limm , n olan , an ) = 0. If everyCauchy

**sequence**is convergent , a metric space is said to be complete . The next...

Page 68

which converges weakly to a point in X. Every

a Cauchy

which converges weakly to a point in X. Every

**sequence**{ Xn } such that { * x } isa Cauchy

**sequence**of scalars for each x * e X * is called a weak Cauchy**sequence**. The space X is said to be weakly complete if every weak Cauchy**sequence**...Page 345

Show that a

and converges at each point of the boundary of D ( to f ) . Show that a subset of A

( D ) ...

Show that a

**sequence**of functions in A ( D ) is a weak Cauchy**sequence**( a**sequence**converging weakly to f in A ( D ) ) if and only if it is uniformly boundedand converges at each point of the boundary of D ( to f ) . Show that a subset of A

( D ) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 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

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero