## Linear Operators: General theory |

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Page 67

23 Theorem. A closed linear manifold in a

Let ?J be a closed linear manifold in the

defined by the equation f (x*)y = x*y, y cty. Clearly \£{x*)\ ^ |**|, so that f : X* -»□?)

*.

23 Theorem. A closed linear manifold in a

**reflexive**B-space is**reflexive**. Proof.Let ?J be a closed linear manifold in the

**reflexive**space X. Let £ : x* -*y* bedefined by the equation f (x*)y = x*y, y cty. Clearly \£{x*)\ ^ |**|, so that f : X* -»□?)

*.

Page 88

conjugate is isometric was proved by Hahn [3; p. 219] who ... He used the term

regular to describe what we, following Lorch [2], have called

of ...

**Reflexivity**. The fact that the natural isomorphism of a fi-space 3E into its secondconjugate is isometric was proved by Hahn [3; p. 219] who ... He used the term

regular to describe what we, following Lorch [2], have called

**reflexive**. Examplesof ...

Page 425

Q.E.D. Theorems 2 and 5 lead to an important result on

Theorem. A B-space is

the weak topology. Proof. Let 3£ be a

...

Q.E.D. Theorems 2 and 5 lead to an important result on

**reflexive**spaces. 7Theorem. A B-space is

**reflexive**if and only if its closed unit sphere is compact inthe weak topology. Proof. Let 3£ be a

**reflexive**.B-spaee, and let x be the natural...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations disjoint Doklady Akad Duke Math element ergodic theorem exists finite dimensional function defined Hausdorff space Hence Hilbert space homeomorphism ibid inequality integral Lebesgue Lebesgue measure Lemma linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc Proof properties proved real numbers reflexive Riesz Russian Sbornik N. S. scalar self-adjoint semi-group sequentially compact Show simple functions spectral Studia Math subset subspace Suppose theory topological space Trans uniformly Univ valued function Vber vector space weak topology weakly compact zero