## Linear Operators: General theory |

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Page 421

It is obvious that y> is a linear functional on the subspace T(X) of By II. 8. 11, it

can be extended to a linear functional ytj on E". By IV. 3. 7, yij has the form n VilVl

• • »n] = i=l

functional ...

It is obvious that y> is a linear functional on the subspace T(X) of By II. 8. 11, it

can be extended to a linear functional ytj on E". By IV. 3. 7, yij has the form n VilVl

• • »n] = i=l

**Hence**n g(t) = 2«(/.(«). *eX. Q.E.D. Proof of Theorem 9. liveryfunctional ...

Page 423

the origin, and

continuous, and t/*e3)*. Then y* T is a linear functional on X which is 3£*-

continuous.

**Hence**\yf(Tx)\ < e , so that TxeN(0;y* y*, e). Therefore, Tis weakly continuous atthe origin, and

**hence**at every point. Conversely, suppose that T is weaklycontinuous, and t/*e3)*. Then y* T is a linear functional on X which is 3£*-

continuous.

Page 441

covering of Q; let {<7,+t-7}, i = 1, . . n, be a finite subeovering. Put Kt — co{(qiJrU)

n Q) Q Qi+U. Then K( is a closed, and

co(Q) = co^i U • • • U A'„) = co(A\ U • • • U Kn), by an easy induction on Lemma 2.5

.

covering of Q; let {<7,+t-7}, i = 1, . . n, be a finite subeovering. Put Kt — co{(qiJrU)

n Q) Q Qi+U. Then K( is a closed, and

**hence**a compact, subset of co(Q).**Hence**co(Q) = co^i U • • • U A'„) = co(A\ U • • • U Kn), by an easy induction on Lemma 2.5

.

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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