## Linear Operators: General theory |

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Page 424

'g *Prz2 — pf-nS'i- Since each projection is a continuous map. each o the sets A(x

. y) and B(x, x) is closed. Hence rAT = Ox ft j.^lx. y) <- Om 4, xf x B(x, x) is also

closed. Q.E.D. 2 Theorem. (Alaoglu) The

space ...

'g *Prz2 — pf-nS'i- Since each projection is a continuous map. each o the sets A(x

. y) and B(x, x) is closed. Hence rAT = Ox ft j.^lx. y) <- Om 4, xf x B(x, x) is also

closed. Q.E.D. 2 Theorem. (Alaoglu) The

**closed unit sphere**in the conjugatespace ...

Page 425

If x is the natural embedding of a B-space X into X**, then xX is JL*-dense in X**.

Proof. The X*-

the

If x is the natural embedding of a B-space X into X**, then xX is JL*-dense in X**.

Proof. The X*-

**closure**of x(X) is a subspace of X**, which, by Theorem 5, containsthe

**unit sphere**of X**. It follows immediately that it contains every point in X**.Page 485

Since the

Lemma 7 and Lemma 1.5.7 that T*S* is compact in the X** topology of X*. Hence

T* is" weakly compact. Conversely, if T* is weakly compact, it follows from Lemma

7 ...

Since the

**closed unit sphere**S* of1?)* is ?)-compact (V.4.2), it follows fromLemma 7 and Lemma 1.5.7 that T*S* is compact in the X** topology of X*. Hence

T* is" weakly compact. Conversely, if T* is weakly compact, it follows from Lemma

7 ...

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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