## Linear Operators, Part 1 |

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Page 26

Thus S ( x , 2 - K ) C Um , which shows that the

metric topology onto X with the given topology is continuous . Hence the

is a homeomorphism and the space is metrizable . Q.E.D. 7. Convergence and ...

Thus S ( x , 2 - K ) C Um , which shows that the

**identity**mapping of X with themetric topology onto X with the given topology is continuous . Hence the

**identity**is a homeomorphism and the space is metrizable . Q.E.D. 7. Convergence and ...

Page 249

Nelson Dunford, Jacob T. Schwartz. PROOF . The

x | 2 + 2 y | ?, X , ye , called the parallelogram

the axioms . If d = inf ( x - k | the preceding

...

Nelson Dunford, Jacob T. Schwartz. PROOF . The

**identity**\ x + y2 + \ x — y | 2 2 \x | 2 + 2 y | ?, X , ye , called the parallelogram

**identity**, follows immediately fromthe axioms . If d = inf ( x - k | the preceding

**identity**shows that keK 10 个| ki - k ; ] 2...

Page 414

Thus , if N is any neighborhood of the

This means that ( N - N + ko ) n ( p - A ) +0 . If M is any neighborhood of the

...

Thus , if N is any neighborhood of the

**identity**, Ky ( N + ko ) n ( N + p - A ) +0 .This means that ( N - N + ko ) n ( p - A ) +0 . If M is any neighborhood of the

**identity**, there is a neighborhood N of the**identity**such that N - NCM . Thus , any...

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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