## Linear Operators: General theory |

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Page 424

Since each projection is a continuous map, each o the sets A(x. y) and B(<x, x) is

closed. Hence rK = n, yt$A(x, y) n nae<(> x(\ B(x. x) is also closed. Q.E.D. 2

Theorem. (Alaoglu) The

space ...

Since each projection is a continuous map, each o the sets A(x. y) and B(<x, x) is

closed. Hence rK = n, yt$A(x, y) n nae<(> x(\ B(x. x) is also closed. Q.E.D. 2

Theorem. (Alaoglu) The

**closed unit sphere**in the conjugate space X* of the B-space ...

Page 458

5 If the

finite number of extremal points, then X is not isometrically isomorphic to the

conjugate of any B-space. 6 Let S be a topological space, and let C(S) be the fi-

space ...

5 If the

**closed unit sphere**of an infinite dimensional B-space X contains only afinite number of extremal points, then X is not isometrically isomorphic to the

conjugate of any B-space. 6 Let S be a topological space, and let C(S) be the fi-

space ...

Page 485

Since the

7 and Lemma 1.5.7 that T*S* is compact in ... If S, S** are the

in X, X**, respectively, and if x is the natural embedding of X into X**, then by ...

Since the

**closed unit sphere**S* of?)* is 7)-compact (V.4.2), it follows from Lemma7 and Lemma 1.5.7 that T*S* is compact in ... If S, S** are the

**closed unit spheres**in X, X**, respectively, and if x is the natural embedding of X into X**, then by ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero