## Linear Operators: General theory |

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Page 50

xG into G. (b) In a linear topological space X, all linear combinations of any

number of scalars xv . . ., Xm, and vectors xv . . ., xn, are ... If X is a linear

topological space, the closure of sp(2?), denoted by sp(5), is called the

xG into G. (b) In a linear topological space X, all linear combinations of any

number of scalars xv . . ., Xm, and vectors xv . . ., xn, are ... If X is a linear

topological space, the closure of sp(2?), denoted by sp(5), is called the

**closed****linear manifold**...Page 249

Two manifolds 9J2, 92 in £ are orthogonal manifolds if (9ft, 91) = 0. We write x J_

y to ... If 9ft is a

Lemma 2, an m e9ft such that \x— m\ = d = infmeSjj|a; — mJ. It will now be ...

Two manifolds 9J2, 92 in £ are orthogonal manifolds if (9ft, 91) = 0. We write x J_

y to ... If 9ft is a

**closed linear manifold**and if x is an arbitrary point in § there is, byLemma 2, an m e9ft such that \x— m\ = d = infmeSjj|a; — mJ. It will now be ...

Page 252

The operator E is the orthogonal projection on the

determined by A. Proof. Let yv ....?/„ be distinct elements of A and let y = 22-i(«- y

*)y* so that (fay i*TM** n \y\2 = ZT-iK*. «r*>I" *nd 0 g |a— j/|2 = Ja?|* — (<r, t/)— (t/,

z)+|y|2 ...

The operator E is the orthogonal projection on the

**closed linear manifold**determined by A. Proof. Let yv ....?/„ be distinct elements of A and let y = 22-i(«- y

*)y* so that (fay i*TM** n \y\2 = ZT-iK*. «r*>I" *nd 0 g |a— j/|2 = Ja?|* — (<r, t/)— (t/,

z)+|y|2 ...

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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