## Linear Operators, Part 1 |

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Page 72

11 ) is a

Hint : Given a Cauchy sequence in X / 3 , define a subsequence for which Xx —

Xx + 1 + 3 ) < 2 - k , k = 1 , 2 , . . . , and show that a Cauchy sequence in X can be

...

11 ) is a

**B**-**space**( or an F - space ) with the metric 1x + 31 = inf ( x + zl . 283 (Hint : Given a Cauchy sequence in X / 3 , define a subsequence for which Xx —

Xx + 1 + 3 ) < 2 - k , k = 1 , 2 , . . . , and show that a Cauchy sequence in X can be

...

Page 89

Then X is isomorphic and isometric with a dense linear subspace of an F - space

Ž . The space ž is uniquely determined up to isometric isomorphism . If X is a

normed linear space , then X is a

as ...

Then X is isomorphic and isometric with a dense linear subspace of an F - space

Ž . The space ž is uniquely determined up to isometric isomorphism . If X is a

normed linear space , then X is a

**B**-**space**. The proof of this theorem proceedsas ...

Page 437

15 If X is a separable linear topological space and A is an Xcompact subset of X *

, then the X - topology of A is a metric topology . 16 If X is a separable

a convex subset A of X * is Xclosed if and only if w * A , and lim x * ( x ) = x * ( x ) ...

15 If X is a separable linear topological space and A is an Xcompact subset of X *

, then the X - topology of A is a metric topology . 16 If X is a separable

**B**-**space**,a convex subset A of X * is Xclosed if and only if w * A , and lim x * ( x ) = x * ( x ) ...

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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