## Linear Operators, Part 1 |

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Page 54

A linear mapping of one F - space into another is continuous if and only if it maps

linear and continuous , and let B C X be

A linear mapping of one F - space into another is continuous if and only if it maps

**bounded**sets into**bounded**sets . Proof . Let X , Y be F - spaces , let T : X + Y belinear and continuous , and let B C X be

**bounded**. For every neighborhood V of ...Page 231

Since this set is clearly closed and since D is connected , \ ( x ) = \ | ( zo ) for all z

in D. Maximum modulus principle for a strip . Let / ( x + iy ) = f ( x ) be an analytic

function with values in a complex B - space X , defined and uniformly

...

Since this set is clearly closed and since D is connected , \ ( x ) = \ | ( zo ) for all z

in D. Maximum modulus principle for a strip . Let / ( x + iy ) = f ( x ) be an analytic

function with values in a complex B - space X , defined and uniformly

**bounded**on...

Page 345

if and only if it is

and each j = 0 , 1 , ... , p . ( c ) Co is not weakly complete , and not reflexive . ( d )

A subset A CCP is conditionally compact if and only if it is

...

if and only if it is

**bounded**and to ( s ) converges ( to f ( s ) ) for each s . in [ a , b ]and each j = 0 , 1 , ... , p . ( c ) Co is not weakly complete , and not reflexive . ( d )

A subset A CCP is conditionally compact if and only if it is

**bounded**and for each...

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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