Linear Operators: General theory |
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Page 113
... integrable if and only if it is v ( u ) -integrable . If f is u - integrable so is f ( · ) . If { f } is a sequence of u - integrable simple functions determining f in accordance with the preced- ing definition , then the sequence ...
... integrable if and only if it is v ( u ) -integrable . If f is u - integrable so is f ( · ) . If { f } is a sequence of u - integrable simple functions determining f in accordance with the preced- ing definition , then the sequence ...
Page 180
... integrable . By Lemma 3 , fg is measur- able . The u - integrability of fg follows from Theorem 2.22 when it is observed ( by Theorem 4 ) that f ( ) g ( ) is u - integrable . Theorem 4 applies in the same manner to prove the 2 - ...
... integrable . By Lemma 3 , fg is measur- able . The u - integrability of fg follows from Theorem 2.22 when it is observed ( by Theorem 4 ) that f ( ) g ( ) is u - integrable . Theorem 4 applies in the same manner to prove the 2 - ...
Page 181
... ( u ) -integrable . The v ( u ) -integrability of g ( ) , however , follows from Theorem 4 just as in the preceding argument . Thus only formula [ * ] remains to be verified . By the argument used in the proof of Corollary 5 it is seen ...
... ( u ) -integrable . The v ( u ) -integrability of g ( ) , however , follows from Theorem 4 just as in the preceding argument . Thus only formula [ * ] remains to be verified . By the argument used in the proof of Corollary 5 it is seen ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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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