## Linear Operators, Part 1 |

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Page 72

16 Let X be a normed linear space which is not assumed to be complete , and let

Z be a closed

also . 17 DEFINITION . If X is a normed linear space , and Z ÇX , the set 2 + = { x ...

16 Let X be a normed linear space which is not assumed to be complete , and let

Z be a closed

**subspace**of X . Then , if Z and X / 3 are complete , show that X isalso . 17 DEFINITION . If X is a normed linear space , and Z ÇX , the set 2 + = { x ...

Page 504

Let X * be the conjugate space of a B - space X . Let MCX * be a separable linear

manifold . Then there exists a closed separable

equivalent to a

...

Let X * be the conjugate space of a B - space X . Let MCX * be a separable linear

manifold . Then there exists a closed separable

**subspace**y of X such that M isequivalent to a

**subspace**of Y * . Proof . Let { w * } be a countable dense subset of...

Page 513

16 If Y is a closed

N is finite dimensional , it does not follow that . Y is closed . 17 Let X be a B -

space and ...

16 If Y is a closed

**subspace**of a B - space and N is a finite dimensional**subspace**, then y o N is a closed**subspace**. If y O N is a closed**subspace**, andN is finite dimensional , it does not follow that . Y is closed . 17 Let X be a B -

space and ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero