## Linear Operators: General theory |

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Page 372

769]

an Euclidean space. This, in particular, implies the completeness. Theorem 3.5,

which characterizes locally compact 5-spaces is due to F. Riesz [4], The "

Schwarz ...

769]

**proved**that any finite dimensional topological linear space is equivalent toan Euclidean space. This, in particular, implies the completeness. Theorem 3.5,

which characterizes locally compact 5-spaces is due to F. Riesz [4], The "

Schwarz ...

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**...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero