## Linear Operators: General theory |

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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 g ( am , 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

is a Cauchy

which converges weakly to a point in X . Every

**sequence**{ xn } such that { x * xx }is a 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 ...Page 345

Show that a

and converges at each point of the boundary of D ( to 1 ) . 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 1 ) . Show that a subset of A

( D ) ...

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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