## Linear Operators: General theory |

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Page 165

If / is a /tj-integrable simple function, it is evident that / is also a //-integrable

simple function, and that J" Ef(s)fa(ds) = J ... It follows from (d) and (e) that a

function / e

spaces is the ...

If / is a /tj-integrable simple function, it is evident that / is also a //-integrable

simple function, and that J" Ef(s)fa(ds) = J ... It follows from (d) and (e) that a

function / e

**LP**(**S**, Ev fa, X) is also in**LP**(**S**, E, ju, X), and that its norm in bothspaces is the ...

Page 286

discussion of the present section covers the space lp as well as the space

,fi). 1 Theorem. If 1 <p < oo and l/p + l/q = 1, there is an isometric isomorphism

between L*(S, E, fi) and LV(S, E, /z) in which corresponding vectors x* and g are

...

discussion of the present section covers the space lp as well as the space

**Lp**(**S**,E,fi). 1 Theorem. If 1 <p < oo and l/p + l/q = 1, there is an isometric isomorphism

between L*(S, E, fi) and LV(S, E, /z) in which corresponding vectors x* and g are

...

Page 663

11, weakly sequentially compact as a sequence in L^S, E, fj,). The desired

conclusion follows from Corollary 3. Q.E.D. The preceding general results will

now be applied to the case of an operator in

— f<p, ...

11, weakly sequentially compact as a sequence in L^S, E, fj,). The desired

conclusion follows from Corollary 3. Q.E.D. The preceding general results will

now be applied to the case of an operator in

**LP**(**S**, E, fi) having the special form Tf— f<p, ...

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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 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