## Linear Operators: General theory |

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Page 60

2\x\ ' £ ' This shows that (i)

of T at 0; so (iv)

finite, then for an arbitrary x =fc 0, This shows that (iii)

2\x\ ' £ ' This shows that (i)

**implies**(iv). Statement (iv) clearly**implies**the continuityof T at 0; so (iv)

**implies**(ii). This (i), (ii), and (iv) are equivalent. If M = sup |Tai| isfinite, then for an arbitrary x =fc 0, This shows that (iii)

**implies**(iv). It is obvious ...Page 280

That (1)

to show that condition (3) of that theorem

that S may be embedded as a dense subset of a compact Hausdorff space St in ...

That (1)

**implies**(2) can be proved in a manner similar to that used in Theorem 14to show that condition (3) of that theorem

**implies**(4). From Corollary 19 it followsthat S may be embedded as a dense subset of a compact Hausdorff space St in ...

Page 394

For example, a partially ordered vector space is a vector space SB in which there

is defined a relation x 22 y for some pairs of elements in SB for which (i) x 2; 0

and — x 2: 0

For example, a partially ordered vector space is a vector space SB in which there

is defined a relation x 22 y for some pairs of elements in SB for which (i) x 2; 0

and — x 2: 0

**imply**x — 0; (ii) x 2r y and y 2; z**imply**x 2; z; (iii) x 2: 0 and X a ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero