## 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 462

116]. Dieudonne [3; p. 109]

induction. The case of Theorem 3.9 in which £ = 3)* and r = s2), was

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

256].

116]. Dieudonne [3; p. 109]

**proved**3.9 after establishing Lemma 3.10 byinduction. The case of Theorem 3.9 in which £ = 3)* and r = s2), was

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

256].

Page 463

124] also

with that of closure in the X topology of X*. Alaoglu [1; p. 256] and Kakutani [2; p.

170] independently established the equivalence of these types of closure without

...

124] also

**proved**that in the case of a separable space these notions coincidewith that of closure in the X topology of X*. Alaoglu [1; p. 256] and Kakutani [2; p.

170] independently established the equivalence of these types of closure without

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact