## Linear Operators: General theory |

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Page 412

The

separates the subsets M—N and {0} of X. The proof is elementary, and is left to

the reader. In dealing with subspaces, it is often convenient to make use of the

following ...

The

**linear functional**f separates the subsets M and N of X if and only if itseparates the subsets M—N and {0} of X. The proof is elementary, and is left to

the reader. In dealing with subspaces, it is often convenient to make use of the

following ...

Page 417

a

has an interior point, then the functional is continuous. Proof. Let X be a linear

topological space, and Av A2 Q 3£. 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 X be a linear

topological space, and Av A2 Q 3£. Let h be a

**linear functional**separating A1 and...

Page 452

If K is a closed cone with vertex p in a real locally convex linear topological space

X, and K ^ X, then there exists a nonzero continuous

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

If K is a closed cone with vertex p in a real locally convex linear topological space

X, and K ^ X, then there exists a nonzero continuous

**linear functional**tangent to Kat p. If A is a subset of X, and p is in A, then there exists a non-zero continuous ...

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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