## Linear Operators: General theory |

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Page 54

Since the

proof is complete . Q.E.D. We now are in a position to prove a number of basic

results about F - spaces . 14 THEOREM . A linear

Since the

**mapping**x + x / ko is a homeomorphism of X with itself , by 10 ( ii ) , theproof is complete . Q.E.D. We now are in a position to prove a number of basic

results about F - spaces . 14 THEOREM . A linear

**mapping**of one F - space into ...Page 55

The Interior

following theorem . 1 THEOREM . Under a continuous linear map of one F -

space onto all of another , the image of every open set is open . PROOF . Let X ,

Y be F ...

The Interior

**Mapping**Principle The interior**mapping**principle is stated in thefollowing theorem . 1 THEOREM . Under a continuous linear map of one F -

space onto all of another , the image of every open set is open . PROOF . Let X ,

Y be F ...

Page 493

Nelson Dunford. 2 * Conversely , if ( a ) and ( b ) are satisfied for the

which sends x * into u ( • ) * , then it follows that for each fixed f e C ( S ) the

therefore ...

Nelson Dunford. 2 * Conversely , if ( a ) and ( b ) are satisfied for the

**mapping**which sends x * into u ( • ) * , then it follows that for each fixed f e C ( S ) the

**mapping**x * → → [ st ( s ) u ( ds ) ** is continuous in the X topology of X * andtherefore ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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