## Linear Operators: General theory |

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Page 91

admits an

invariant metric and is complete under each invariant metric. Thus every

complete linear metric space can be metrized to be an F-space. Further, a

normed linear space ...

admits an

**equivalent**metric under which it is complete, then G admits aninvariant metric and is complete under each invariant metric. Thus every

complete linear metric space can be metrized to be an F-space. Further, a

normed linear space ...

Page 347

is

measure {En} in E such that every measurable subset of S— \JTM=1E„ is either

an atom of infinite measure or a null set. 50 Show that no space

is

**equivalent**to lx if and only if there exists a countable collection of atoms of finitemeasure {En} in E such that every measurable subset of S— \JTM=1E„ is either

an atom of infinite measure or a null set. 50 Show that no space

**equivalent**to a ...Page 663

... discover conditions on <p which are sufficient to insure that the operator T

maps LP(S, E, ft) into E, fi) and has the corresponding sequence {A{n)} bounded.

Since the points in LP(S, E, fi) are not functions but classes of

functions, ...

... discover conditions on <p which are sufficient to insure that the operator T

maps LP(S, E, ft) into E, fi) and has the corresponding sequence {A{n)} bounded.

Since the points in LP(S, E, fi) are not functions but classes of

**equivalent**functions, ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### 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 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