Linear Operators: General theory |
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... mean that A is a proper subset of B. The complement of a set A relative to a set B is the set whose elements are in B ... means that the following distributive laws hold : acA ... xU_a = U ̧ ( xa ) , × U ( Na ) = ~ ( x ( a ) . aεA aεA a ...
... mean that A is a proper subset of B. The complement of a set A relative to a set B is the set whose elements are in B ... means that the following distributive laws hold : acA ... xU_a = U ̧ ( xa ) , × U ( Na ) = ~ ( x ( a ) . aεA aεA a ...
Page 381
... means of a countably additive measure if and only if any one of the following conditions hold : ( 1 ) fne C ( S ) , n = 0 , 1 , 2 , . . . and fn ( s ) ↑ fo ( s ) , se S imply that Info uniformly on S ; ( 2 ) every continuous real ...
... means of a countably additive measure if and only if any one of the following conditions hold : ( 1 ) fne C ( S ) , n = 0 , 1 , 2 , . . . and fn ( s ) ↑ fo ( s ) , se S imply that Info uniformly on S ; ( 2 ) every continuous real ...
Page 660
... mean and pointwise convergence theorems for the contin- uous case . Section 8 is concerned with a certain class of operators T for which the sequence of averages { N - 1 NT " } converges in the uniform topology of operators . Finally ...
... mean and pointwise convergence theorems for the contin- uous case . Section 8 is concerned with a certain class of operators T for which the sequence of averages { N - 1 NT " } converges in the uniform topology of operators . Finally ...
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