## Linear Operators: General theory |

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Page 421

/-continuous, by Lemma 8. Conversely, let g 0 be a linear functional on X which is

/"-continuous. There exists a /-neighborhood N(0; fv . . ., /„; e) which is mapped ...

**Hence**n g(x) = 2 «i /<(«)» -ceX- Q.E.D. Proof of Theorem 9. Every functional in r is/-continuous, by Lemma 8. Conversely, let g 0 be a linear functional on X which is

/"-continuous. There exists a /-neighborhood N(0; fv . . ., /„; e) which is mapped ...

Page 423

Then \x*\ <\y*\- \T\, and so x* « X*. If x e N(0; x*, . . ., x*. s) then \x*(x)\ < e.

*{Tx)\ < e, so that TxeN(0; y*, . . ., y*, e). Therefore, T is weakly continuous at the

origin, and

...

Then \x*\ <\y*\- \T\, and so x* « X*. If x e N(0; x*, . . ., x*. s) then \x*(x)\ < e.

**Hence**\y*{Tx)\ < e, so that TxeN(0; y*, . . ., y*, e). Therefore, T is weakly continuous at the

origin, and

**hence**atevery point. Conversely, suppose that T is weakly continuous...

Page 441

Put K( = co((<7,-+l/)n Q) Q (ji+L - Then A',, is a closed, and

subset of co(Q).

induction on Lemma 2.5. It follows readily that p has the form p = 2"«.iai^» at ...

Put K( = co((<7,-+l/)n Q) Q (ji+L - Then A',, is a closed, and

**hence**a compact,subset of co(Q).

**Hence**co(Q) = co(Ar! u • • • U Kn) = co(A'j u . . . U Kn), by an easyinduction on Lemma 2.5. It follows readily that p has the form p = 2"«.iai^» at ...

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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