Linear Operators: General theory |
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Page 119
... consider a function f ( vector or extended real - valued ) which is defined only on the complement of a u - null set ... considering this somewhat more extended class of functions we make no change in F ( S , Z , u , X ) , or in any of ...
... consider a function f ( vector or extended real - valued ) which is defined only on the complement of a u - null set ... considering this somewhat more extended class of functions we make no change in F ( S , Z , u , X ) , or in any of ...
Page 139
... consider intervals I having one of the two forms a , d or ( c , d ] where a < c < db . For such intervals , place μ ( [ a , d ] ) = d - a , u ( ( c , d ] ) = d - c . Let consist of all finite unions of such intervals . It is clear that ...
... consider intervals I having one of the two forms a , d or ( c , d ] where a < c < db . For such intervals , place μ ( [ a , d ] ) = d - a , u ( ( c , d ] ) = d - c . Let consist of all finite unions of such intervals . It is clear that ...
Page 468
... consider real Euclidean space . Further , the Weierstrass approximation theorem for continuous functions of n variables implies that every continuous map of S into itself is the uniform limit of a sequence { } of infinite- ly ...
... consider real Euclidean space . Further , the Weierstrass approximation theorem for continuous functions of n variables implies that every continuous map of S into itself is the uniform limit of a sequence { } of infinite- ly ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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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