## Linear Operators: General theory |

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Page 373

This generalizes and abstracts a result

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 (

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 4.4) was**proved**...Page 385

We now comment briefly on the Theorems 6.18 — 6.27. 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

...

We now comment briefly on the Theorems 6.18 — 6.27. 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**...

Page 462

116]. Dieudonn<5 [8; p. 109]

induction. The case of Theorem 3.9 in which X = ¥)* and r = SI), was

separable case by Banach [1; p. 131] and in full generality by Alaoglu [1; p. 2561.

116]. Dieudonn<5 [8; p. 109]

**proved**3.9 after establishing Lemma 3.10 byinduction. The case of Theorem 3.9 in which X = ¥)* and r = SI), was

**proved**in theseparable case by Banach [1; p. 131] and in full generality by Alaoglu [1; p. 2561.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero