Linear Operators: General theory |
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... respectively . Finally if z is a complex number and 2 = xiy , where x and y are real , then x and y are called the real part and the imaginary part of ≈ and are denoted by A ( z ) and I ( z ) , respectively . 2. Partially Ordered ...
... respectively . Finally if z is a complex number and 2 = xiy , where x and y are real , then x and y are called the real part and the imaginary part of ≈ and are denoted by A ( z ) and I ( z ) , respectively . 2. Partially Ordered ...
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 ...
... 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 ...
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
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ