Linear Operators: General theory |
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Page 147
... vanishes almost everywhere . If f is μ - integrable then [ f ( s ) μ ( ds ) = 0 for every E in Σ if and only if f vanishes almost everywhere . PROOF . It is clear that a function vanishing almost everywhere is a null function ...
... vanishes almost everywhere . If f is μ - integrable then [ f ( s ) μ ( ds ) = 0 for every E in Σ if and only if f vanishes almost everywhere . 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 , E , u ) ... vanish outside S , and obtain an h , e L ( S , E , u ) . Now for u - almost all se S , n === the sequence { h , ( s ) ...
... vanishes u - almost every- where outside of a set of o - finite measure . We may therefore suppose that ( S , E , u ) ... vanish outside S , and obtain an h , e L ( S , E , u ) . Now for u - almost all se S , n === the sequence { h , ( s ) ...
Page 572
... vanishes identically on S ( α ,, ɛ ( α ; ) ) . 1 = Thus , if U1 is the union of those spheres S ( α1 , ɛ ( α ; ) ) which contain an infinite number of points of o ( T ) , then ƒ vanishes identi- cally on U1 . Hence , U , contains all ...
... vanishes identically on S ( α ,, ɛ ( α ; ) ) . 1 = Thus , if U1 is the union of those spheres S ( α1 , ɛ ( α ; ) ) which contain an infinite number of points of o ( T ) , then ƒ vanishes identi- cally on U1 . Hence , U , contains all ...
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