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 , Z , u ) ... vanish outside S and obtain an h , e L∞ ( S , 2 , μ ) . Now for u - almost all se S , 00 the sequence { h , ( s ) ...
... vanishes u - almost every- where outside of a set of o - finite measure . We may therefore suppose that ( S , Z , u ) ... vanish outside S and obtain an h , e L∞ ( S , 2 , μ ) . Now for u - almost all se S , 00 the sequence { h , ( s ) ...
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
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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 compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 theory topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ