## Linear Operators: General theory |

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Page 244

Finite Dimensional Spares The space En, as will be seen presently, is the

prototype of all n-dimensional normed linear spaces, and

observed first that En is a B-space. The Minkowski inequality (III. 3. 3) shows E" to

be a ...

Finite Dimensional Spares The space En, as will be seen presently, is the

prototype of all n-dimensional normed linear spaces, and

**hence**it should beobserved first that En is a B-space. The Minkowski inequality (III. 3. 3) shows E" to

be a ...

Page 423

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

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

origin, and

continuous ...

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

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

origin, and

**hence**at every point. Conversely, suppose that T is weaklycontinuous ...

Page 441

Put Kt = co((91+C)n Q) Q + Then A', is a closed, and

co(Q).

on Lemma 2.5. It follows readily that p has the form p = 2"=1 a(k„ o, Si 0, ]£"=1 a, ...

Put Kt = co((91+C)n Q) Q + Then A', is a closed, and

**hence**a compact, subset ofco(Q).

**Hence**c3(Q) = co(A\ u • • • U Kn) = co(A'j u . • • U AJ, by an easy inductionon Lemma 2.5. It follows readily that p has the form p = 2"=1 a(k„ o, Si 0, ]£"=1 a, ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero