## 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 418

p and q are distinct points of a locally convex linear topological space X, there is

a continuous

p and q are distinct points of a locally convex linear topological space X, there is

a continuous

**linear functional**f defined ... T2) have the same continuous**linear****functionals**, then a convex set is closed in (X, rx) if and only if it is closed in (X, t2).Page 452

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

is not dense in X. Proof. If qiK, 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 qiK, 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 | 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 closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero