## Linear Operators: General theory |

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Page 167

X) is isometrically equivalent to the set of all functions. in

outside E, so that the former space may be regarded as a subspace of the latter.

3 Lemma. Let (S,E ,fi) be a positive measure space and fa the restriction of p to a

...

X) is isometrically equivalent to the set of all functions. in

**LP**(**S**, E, fi, X) vanishingoutside E, so that the former space may be regarded as a subspace of the latter.

3 Lemma. Let (S,E ,fi) be a positive measure space and fa the restriction of p to a

...

Page 298

Since |C7„| g 1 it follows by III.3.8 that UJ ->/ for all / in

oo, the theorem follows from Lemma IV. 5. 4. If fi(S) < oo a similar argument

proves the desired result in L^(S, Z, /x). Q.E.D. If S is n-dimensional Euclidean

space, ...

Since |C7„| g 1 it follows by III.3.8 that UJ ->/ for all / in

**LP**(**S**, Z, fi). Hence if 1 p <oo, the theorem follows from Lemma IV. 5. 4. If fi(S) < oo a similar argument

proves the desired result in L^(S, Z, /x). Q.E.D. If S is n-dimensional Euclidean

space, ...

Page 336

Q.E.D. The natural ordering just used for functions in M(S, E, p) has already been

employed for functions in the spaces

8.23 we proved that if a subset of one of these spaces is bounded above by ...

Q.E.D. The natural ordering just used for functions in M(S, E, p) has already been

employed for functions in the spaces

**Lp**(**S**,E.p), 1 p 5S oo. In theorems 8.22 and8.23 we proved that if a subset of one of these spaces is bounded above by ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

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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