## Linear Operators: General theory |

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Page 57

A continuous

continuous

one-to- one

A continuous

**linear**one-to-one**map**of one F-space onto all of another has acontinuous

**linear**inverse. Proof. Let X, ?) be F-spaces and T a continuous**linear**one-to- one

**map**with TX = 3). Since (T-1)-1 = T**maps**open sets onto open sets ...Page 58

Let rv r2 be metric topologies in the linear space X for which the spaces Xj = (X, t^

), X2 = (X, t2) are F-spaces. If ^ C t2, then the one-to-one

onto Xx is continuous. By Theorem 2, it is a homeomorphism, and so x1 = t2.

Let rv r2 be metric topologies in the linear space X for which the spaces Xj = (X, t^

), X2 = (X, t2) are F-spaces. If ^ C t2, then the one-to-one

**linear map**x -*□ x of X2onto Xx is continuous. By Theorem 2, it is a homeomorphism, and so x1 = t2.

Page 490

For example, while it is easy to see that the general continuous

„[0, 1], p > 1, to L„[0, 1] has the form d f1 g(s) = - K(s, t)f(t)dt, as Jo no satisfactory

expression for the norm of T is known. No conditions on K(s, t) are known which ...

For example, while it is easy to see that the general continuous

**linear map**from L„[0, 1], p > 1, to L„[0, 1] has the form d f1 g(s) = - K(s, t)f(t)dt, as Jo no satisfactory

expression for the norm of T is known. No conditions on K(s, t) are known which ...

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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