## Linear Operators: General theory |

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Page 19

The metric topology in .ST is the smallest topology containing the spheres. The

set X, with its metric topology, is called a

such that there exists a metric function whose topology is the same as the original

...

The metric topology in .ST is the smallest topology containing the spheres. The

set X, with its metric topology, is called a

**metric space**. If X is a topological spacesuch that there exists a metric function whose topology is the same as the original

...

Page 20

A sequence {an} in a

every Cauchy sequence is convergent, a

The next three lemmas are immediate consequences of the definitions. 6 Lemma

...

A sequence {an} in a

**metric space**is a Cauchy sequence if limm „ g{am, a„) = 0. Ifevery Cauchy sequence is convergent, a

**metric space**is said to be complete.The next three lemmas are immediate consequences of the definitions. 6 Lemma

...

Page 459

13 Let X be a J3-space, and let K* be a bounded X-closed convex subset of X*.

Show that K* has a continuous tangent functional at each point of a dense subset

of its boundary. 14 Definition. A point p of a subset A of a

13 Let X be a J3-space, and let K* be a bounded X-closed convex subset of X*.

Show that K* has a continuous tangent functional at each point of a dense subset

of its boundary. 14 Definition. A point p of a subset A of a

**metric space**is called ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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