Linear Operators: General theory |
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Page 4
... respectively . Finally if z is a complex number and 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 ( ≈ ) , respectively . = 2. Partially Ordered ...
... respectively . Finally if z is a complex number and 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 ( ≈ ) , 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 ** ( × S ) = 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 ** ( × S ) = x ( TS ) x ( TS ) , where S1 is ...
Page 666
... respectively when they are considered as operating in L , L ( S , E , u ) . Likewise for a u - measurable function f , the symbol If is used for ( fsf ( s ) Pμ ( ds ) ) 1 / whether or not this is finite . We shall first prove the ...
... respectively when they are considered as operating in L , L ( S , E , u ) . Likewise for a u - measurable function f , the symbol If is used for ( fsf ( s ) Pμ ( ds ) ) 1 / whether or not this is finite . We shall first prove the ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ