## Linear Operators: General theory |

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Page 412

Let f be a

/(?)) is not the whole field of scalars, then fffl = o. Proof. Suppose that there exists

a y e "2) with f(y) ^ 0. Then f(a.y/f(y)) = a, so that every scalar is in /(?)). Q.E.D. 12 ...

Let f be a

**linear functional**on the vector space X, and let ?) be a subspace of X. ///(?)) is not the whole field of scalars, then fffl = o. Proof. Suppose that there exists

a y e "2) with f(y) ^ 0. Then f(a.y/f(y)) = a, so that every scalar is in /(?)). Q.E.D. 12 ...

Page 417

a

lias an interior point, tfien the functional is continudus. Proof. Let £ 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 whichlias an interior point, tfien the functional is continudus. Proof. Let £ be a linear

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

**linear functional**separating Ar and...

Page 452

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

X, and K =yt X, then there exists a nonzero continuous

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

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

X, and K =yt X, then there exists a nonzero continuous

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

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