## Linear Operators: General theory |

### From inside the book

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Page 417

a

has an interior point, then the functional is

topological space, and Av A2 Q X. Let h be a

...

a

**linear functional**on a linear topological space separates two sets, one of whichhas an interior point, then the functional is

**continuous**. Proof. Let £ be a lineartopological space, and Av A2 Q X. Let h be a

**linear functional**separating Ax and...

Page 418

Kv K2 are disjoint closed, convex subsets of a locally convex linear topological

space X. and if Aj is compact, tlien some non-zero

X separates Ax and K2. 12 Corollary. // A is a closed convex subset of a locally ...

Kv K2 are disjoint closed, convex subsets of a locally convex linear topological

space X. and if Aj is compact, tlien some non-zero

**continuous linear functional**onX separates Ax and K2. 12 Corollary. // A is a closed convex subset of a locally ...

Page 452

If A is a subset of X, and p is in A, then there exists a non-zero

is not dense in X. Proof. If q J K, then, by 2.12 we can find a functional / and a real

...

If A is a subset of X, and p is in A, then there exists a non-zero

**continuous linear****functional**tangent to A at p if and only if the cone B with vertex p generated by Ais not dense in X. Proof. If q J K, then, by 2.12 we can find a functional / and a real

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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### Common terms and phrases

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