## Linear Operators: General theory |

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Page 452

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

X, and £ # J, then there exists a

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

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

X, and £ # J, then there exists a

**nonzero**continuous linear junctional tangent to Kat p. If A is a subset of X, and p is in A, then there exists a

**non**-**zero**continuous ...Page 557

zero polynomial Sx such that S1(T)x1 = 0. In the same way, there exist

polynomials S{, i = 2, . . ., k such that Si(T)xi = 0. If R = iSj • S2 - • -Sk, then R(T)xt =

0, and consequently R(T)x = 0 for all a;ej. Thus a

zero polynomial Sx such that S1(T)x1 = 0. In the same way, there exist

**non**-**zero**polynomials S{, i = 2, . . ., k such that Si(T)xi = 0. If R = iSj • S2 - • -Sk, then R(T)xt =

0, and consequently R(T)x = 0 for all a;ej. Thus a

**non**-**zero**polynomial R exists ...Page 579

Every

such a number X, the projection E(X) has a

given by the formula where v is the order of the pole. Proof. Since a(T) is compact

, the ...

Every

**non**-**zero**number in a(T) is a pole of T and has finite positive index. Forsuch a number X, the projection E(X) has a

**non**-**zero**finite dimensional rangegiven by the formula where v is the order of the pole. Proof. Since a(T) is compact

, the ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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