## Linear Operators: General theory |

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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 ÇX 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 ÇX be

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

Since this set is clearly closed and since D is connected , \ / ( x ) ) = \ | ( 20 ) for all

z in D . Maximum modulus principle for a strip . Let f ( x + iy ) = | ( 2 ) 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 ) ) = \ | ( 20 ) for all

z in D . Maximum modulus principle for a strip . Let f ( x + iy ) = | ( 2 ) be an

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

**bounded**...Page 345

if and only if it is

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

d ) A subset A CC ” is conditionally compact if and only if it is

if and only if it is

**bounded**and too ! ( s ) converges ( to f ( 1 ) ( 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 CC ” is conditionally compact if and only if it is

**bounded**and for ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero