## Linear Operators: General theory |

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Page 58

The map prx : [ w , Tx ] → x of G onto X is one - to - one ,

.8.3 ) . Hence , by Theorem 2 , its inverse prą ' is

The map prx : [ w , Tx ] → x of G onto X is one - to - one ,

**linear**, and**continuous**( I.8.3 ) . Hence , by Theorem 2 , its inverse prą ' is

**continuous**. Thus T = pry prz ' is**continuous**( I.4.17 ) . Q.E.D. 5 THEOREM . If a**linear**space is an F - space ...Page 417

If a

which has an interior point , then the functional is

a

separating ...

If a

**linear**functional on a**linear**topological space separates two sets , one ofwhich has an interior point , then the functional is

**continuous**. PROOF . Let X bea

**linear**topological space , and A1 , A , C X . Let h be a**linear**functionalseparating ...

Page 418

If Kį , K , are disjoint closed , convex subsets of a locally convex linear topological

space X , and if K , is compact , then some non - zero

on X separates K , and Ką . 12 COROLLARY . If K is a closed convex subset of ...

If Kį , K , are disjoint closed , convex subsets of a locally convex linear topological

space X , and if K , is compact , then some non - zero

**continuous linear**functionalon X separates K , and Ką . 12 COROLLARY . If K is a closed convex subset of ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

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### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero