## 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. 36]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, E, n) if and only if it converges weakly and \fn\ -> |/|. This

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

proved ...

1363] and F.

**Riesz**[13]. Theorem. If \ < p < oo then a sequence {/„} convergesstrongly to f in LP(S, E, n) if and only if it converges weakly and \fn\ -> |/|. This

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

proved ...

Page 609

For an exposition of this method, see

[5; Chap. 11]. A similar treatment is given by Banach [1; Chap. 10]. The line of

argument given here is closer to Nagumo [1] and is a special case of some

results ...

For an exposition of this method, see

**Riesz**and Sz.- Nagy [1; Chap. 4] or Zaanen[5; Chap. 11]. A similar treatment is given by Banach [1; Chap. 10]. The line of

argument given here is closer to Nagumo [1] and is a special case of some

results ...

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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