## Linear Operators: General theory |

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Page 65

14

x*\ = 1 and x*x = \x\. Proof. Apply Lemma 12 with 2J = 0. The x* required in the

present

14

**Corollary**. For every x ^ 0 in a normed linear space X, there is an x* e X* with \x*\ = 1 and x*x = \x\. Proof. Apply Lemma 12 with 2J = 0. The x* required in the

present

**corollary**may then be defined as \x\ times the x* whose existence is ...Page 188

Q.E.D. As in the case of finite measure spaces we shall call the measure space (

S, E, fi) constructed in

-product measure space and write (S, E, fi) = Pit US* Et.fr). The best known ...

Q.E.D. As in the case of finite measure spaces we shall call the measure space (

S, E, fi) constructed in

**Corollary**6 from the cr-finite measure spaces (S(, Zt, [if) the-product measure space and write (S, E, fi) = Pit US* Et.fr). The best known ...

Page 662

2

the manifold {x\Tx = x) of fixed points of T. The complementary projection has the

closure of (I—T)H for its range. Proof. Since (I-T)A(n) = (I—Tn)jn, the identity ...

2

**Corollary**. When the strong limit E = limn A(n) exists it is a projection of H uponthe manifold {x\Tx = x) of fixed points of T. The complementary projection has the

closure of (I—T)H for its range. Proof. Since (I-T)A(n) = (I—Tn)jn, the identity ...

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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 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