## Linear Operators: General theory |

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Page 244

Finite Dimensional Spaces 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 Spaces 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 441

covering of Q; let {g, □+[/}, i = 1, . . ., n, be a finite subcovering. Put Kf = «)((</, +E/)

n Q) Q Qi+U. Then Jf4 is a closed, and

co(Q) = co^ u • • • U Kn) = co^ u • • • U Kn), by an easy induction on Lemma 2.5.

covering of Q; let {g, □+[/}, i = 1, . . ., n, be a finite subcovering. Put Kf = «)((</, +E/)

n Q) Q Qi+U. Then Jf4 is a closed, and

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

Page 485

Lemma 7 that T** is continuous relative to the X*, ?)*** topologies in X**, 3)**,

respectively. If S, S** are the closed unit spheres in X, X**, respectively, and if x is

...

**Hence**T* is weakly compact. Conversely, if T* is weakly compact, it follows fromLemma 7 that T** is continuous relative to the X*, ?)*** topologies in X**, 3)**,

respectively. If S, S** are the closed unit spheres in X, X**, respectively, and if x is

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

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a-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero