Linear Operators: General theory |
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Page 95
... function , complex function , etc. , where the adjectives real and complex refer to the range of the func- tion , the term set function is commonly used in ... DEFINITION . A set function μ defined on a 95 Integration and Set Functions.
... function , complex function , etc. , where the adjectives real and complex refer to the range of the func- tion , the term set function is commonly used in ... DEFINITION . A set function μ defined on a 95 Integration and Set Functions.
Page 182
... function f of Theorems 2 and 7 is called the Radon - Nikodým derivative of 2 with respect to u and is often denoted by dλ / du . Thus dλ du is defined ... function us on the equation ( p - 1 ( E ) ) = μ2 ( E ) defines an additive Σ set ...
... function f of Theorems 2 and 7 is called the Radon - Nikodým derivative of 2 with respect to u and is often denoted by dλ / du . Thus dλ du is defined ... function us on the equation ( p - 1 ( E ) ) = μ2 ( E ) defines an additive Σ set ...
Page 520
... functions defined on a convex set CC Ek . Then the function M defined by is a convex function . M ( x ) = sup M2 ( x ) π PROOF . For each л we have M „ ( xu + ( 1 − a ) v ) ≤ aM „ ( u ) + ( 1 − a ) M , ( v ) , whenever u , ve C , and ...
... functions defined on a convex set CC Ek . Then the function M defined by is a convex function . M ( x ) = sup M2 ( x ) π PROOF . For each л we have M „ ( xu + ( 1 − a ) v ) ≤ aM „ ( u ) + ( 1 − a ) M , ( v ) , whenever u , ve C , and ...
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