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
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