## Linear Operators: General theory |

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Page 373

44] and for integrals by F.

and was also used by Sz.-Nagy [5]. A similar argument may be applied to obtain

this result in any uniformly convex B-space. This generalizes and abstracts a ...

44] and for integrals by F.

**Riesz**[2; p. 456]. Lemma 4.2 is due to F.**Riesz**[8; p. 86]and was also used by Sz.-Nagy [5]. A similar argument may be applied to obtain

this result in any uniformly convex B-space. This generalizes and abstracts a ...

Page 388

1363] and F.

strongly to f in Lp(S,£,/.i) if and only if it converges weakly and |/„| ->• |/|. This

theorem remains valid in any uniformly convex B-space. Theorem 8.15 was

proved ...

1363] and F.

**Riesz**[13]. Theorem. If I < p < oo then a sequence {/„} convergesstrongly to f in Lp(S,£,/.i) if and only if it converges weakly and |/„| ->• |/|. This

theorem remains valid in any uniformly convex B-space. Theorem 8.15 was

proved ...

Page 729

proofs, valid in uniformly convex spaces, were given by G. Birkhoff [7] and F.

contraction in Hilbert space is also a fixed point for its adjoint, was given by

...

proofs, valid in uniformly convex spaces, were given by G. Birkhoff [7] and F.

**Riesz**[16, 18]. Another proof, based on the interesting fact that a fixed point for acontraction in Hilbert space is also a fixed point for its adjoint, was given by

**Riesz**...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 other sections not shown

### 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-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero