## Linear Operators: General theory |

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Page 373

This generalizes and abstracts a result

This generalizes and abstracts a result

**proved**for closed linear manifolds in L2 [ 0 , 1 ] by E. Fischer [ 2 ] . The fact that a linear manifold which is not dense in the entire space has a non - zero orthogonal complement (**proved**in ...Page 385

They are essentially due , at least in the real case , to Stone [ 1 ] , although his terminology and proofs often differ from that given here . It should be mentioned that Theorem 6.22 was

They are essentially due , at least in the real case , to Stone [ 1 ] , although his terminology and proofs often differ from that given here . It should be mentioned that Theorem 6.22 was

**proved**independently by Cech ( 1 ) only ...Page 462

109 ]

109 ]

**proved**3.9 after establishing Lemma 3.10 by induction . The case of Theorem 3.9 in which X Y * and I ' = Y , was**proved**in the separable case by Banach ( 1 ; p . 131 ] and in full generality by Alaoglu ( 1 ; p . 256 ] .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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 metric 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 u-measurable uniform uniformly unique unit valued vector weak weakly compact zero