Linear Operators: General theory |
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Page 421
... Hence .... Yn ] = Σ α ; Yi i - 1 Q.E.D. g ( x ) = n Σaiti ( x ) , i = 1 x € X. PROOF OF THEOREM 9. Every functional ... hence g ( x ) < 1. Since 1 , is a linear space , it contains ma 。 for every integer m . Hence m \ g ( x ) | = | g ...
... Hence .... Yn ] = Σ α ; Yi i - 1 Q.E.D. g ( x ) = n Σaiti ( x ) , i = 1 x € X. PROOF OF THEOREM 9. Every functional ... hence g ( x ) < 1. Since 1 , is a linear space , it contains ma 。 for every integer m . Hence m \ g ( x ) | = | g ...
Page 423
... Hence y * ( Tx ) < e , so that Ta e N ( 0 ; y * , . . . , y * , ε ) . Therefore , Tis weakly continuous at the origin , and hence at every point . Conversely , suppose that T is weakly continuous , and y * e * . Then y * T is a linear ...
... Hence y * ( Tx ) < e , so that Ta e N ( 0 ; y * , . . . , y * , ε ) . Therefore , Tis weakly continuous at the origin , and hence at every point . Conversely , suppose that T is weakly continuous , and y * e * . Then y * T is a linear ...
Page 485
... Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the X * , Y *** topologies in X ** , Y ** , respectively . If S , S ** are the closed unit spheres in X ...
... Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the X * , Y *** topologies in X ** , Y ** , respectively . If S , S ** are the closed unit spheres in X ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ