## Linear Operators: General theory |

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Page 161

u e ba . For a > 0 choose ne so that lun - - Mm Se for m , n 2 ne . Then u ( E ) - Mn

( E ) = limm + ( Um ( E ) - Mn ( E ) ) , from which it follows that lu - unl Se for n z nic

. Hence Plin → Ml , which proves that

u e ba . For a > 0 choose ne so that lun - - Mm Se for m , n 2 ne . Then u ( E ) - Mn

( E ) = limm + ( Um ( E ) - Mn ( E ) ) , from which it follows that lu - unl Se for n z nic

. Hence Plin → Ml , which proves that

**ba**(**S**, E , X ) is complete . It follows ...Page 311

Q . E . D . Next we turn to an investigation of the space

The space

) is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of ...

Q . E . D . Next we turn to an investigation of the space

**ba**(**S**, E ) . 9 THEOREM .The space

**ba**(**S**, E ) is weakly complete . If S is a topological space , the rba ( S) is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of ...

Page 340

16 Let S be a completely regular topological space . Show that C ( S ) is

separable if and only if S is compact and metric . 17 Show that a sequence { 2n }

of elements of

only if there ...

16 Let S be a completely regular topological space . Show that C ( S ) is

separable if and only if S is compact and metric . 17 Show that a sequence { 2n }

of elements of

**ba**(**S**, E ) converge weakly to an element de**ba**(**S**, E ) if andonly if there ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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 |

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