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 u 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 u 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 μ and is often denoted by d2 / du . Thus d2 / du is defined ... function μ on E the equation u1 ( −1 ( E ) ) = μ2 ( E ) defines an additive set ...
... function f of Theorems 2 and 7 is called the Radon - Nikodým derivative of 2 with respect to μ and is often denoted by d2 / du . Thus d2 / du is defined ... function μ on E the equation u1 ( −1 ( E ) ) = μ2 ( E ) defines an additive set ...
Page 520
... functions defined on a convex set CC E. Then the function M defined by is a convex function . M ( x ) = sup M2 ( x ) PROOF . For each л we have Π M „ ( xu + ( 1 — α ) v ) ≤ aM „ ( u ) + ( 1 — α ) M „ ( v ) , whenever u , ve C , and 0 ...
... functions defined on a convex set CC E. Then the function M defined by is a convex function . M ( x ) = sup M2 ( x ) PROOF . For each л we have Π M „ ( xu + ( 1 — α ) v ) ≤ aM „ ( u ) + ( 1 — α ) M „ ( v ) , whenever u , ve C , and 0 ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ