Linear Operators: General theory |
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Page 147
... vanishes almost everywhere . If f is μ - integrable then † ( s ) u ( ds ) = 0 for every E in Σ if and only if f vanishes almost everywhere . E PROOF . It is clear that a function vanishing almost everywhere is a null function ...
... vanishes almost everywhere . If f is μ - integrable then † ( s ) u ( ds ) = 0 for every E in Σ if and only if f vanishes almost everywhere . E PROOF . It is clear that a function vanishing almost everywhere is a null function ...
Page 302
... vanishes u - almost every- where outside of a set of o - finite measure . We may therefore suppose that ( S , 2 , μ ) ... vanish outside S , and obtain an h , e L ( S , E , u ) . Now for u - almost all se S , the sequence { h , ( s ) } is ...
... vanishes u - almost every- where outside of a set of o - finite measure . We may therefore suppose that ( S , 2 , μ ) ... vanish outside S , and obtain an h , e L ( S , E , u ) . Now for u - almost all se S , the sequence { h , ( s ) } is ...
Page 593
... vanishes at certain points of o ( T ) . The next theorem is concerned with this possibility . 1 THEOREM . Let f , fn be in F ( T ) , and let { f ( T ) fn ( T ) } converge to zero in the uniform operator topology . Let f vanish at a ...
... vanishes at certain points of o ( T ) . The next theorem is concerned with this possibility . 1 THEOREM . Let f , fn be in F ( T ) , and let { f ( T ) fn ( T ) } converge to zero in the uniform operator topology . Let f vanish at a ...
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