Linear Operators: General theory |
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Page 483
... respectively . Note that T ** is continuous with the X * , Y * topologies in *** , ** , respectively ( cf. 2.3 ) . Hence , since T ** is an extension of T ( cf. 2.6 ) , ( i ) T ** ( S1 ) T ** ( xS ) = x ( TS ) x ( TS ) , where S1 is the ...
... respectively . Note that T ** is continuous with the X * , Y * topologies in *** , ** , respectively ( cf. 2.3 ) . Hence , since T ** is an extension of T ( cf. 2.6 ) , ( i ) T ** ( S1 ) T ** ( xS ) = x ( TS ) x ( TS ) , where S1 is the ...
Page 484
... respectively . The following result shows that , if T is weakly compact , its adjoint T * has a stronger continuity property . 7 LEMMA . An operator in B ( X , Y ) is weakly compact if and only if its adjoint is continuous with respect ...
... respectively . The following result shows that , if T is weakly compact , its adjoint T * has a stronger continuity property . 7 LEMMA . An operator in B ( X , Y ) is weakly compact if and only if its adjoint is continuous with respect ...
Page 485
... respectively . If S , S ** are the closed unit spheres in X , X ** , respectively , and if x is the natural embedding of X into X ** , then by Theorem V.4.5 , xS is X * -dense in S ** , and so , from the continuity of T ** we see that T ...
... respectively . If S , S ** are the closed unit spheres in X , X ** , respectively , and if x is the natural embedding of X into X ** , then by Theorem V.4.5 , xS is X * -dense in S ** , and so , from the continuity of T ** we see that T ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ