Linear Operators: General theory |
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Page 95
... Functions In contrast to the terms real 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 mathematics for a function whose domain ...
... Functions In contrast to the terms real 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 mathematics for a function whose domain ...
Page 103
... function or , when u is understood , simply a null function if the set S ( f > a ) is a u - null set for each a > 0 . It is important to observe that a null function with respect to a finitely additive set function need not vanish ...
... function or , when u is understood , simply a null function if the set S ( f > a ) is a u - null set for each a > 0 . It is important to observe that a null function with respect to a finitely additive set function need not vanish ...
Page 165
... function , it is evident that ƒ is also a u - integrable simple function , and that SÅ ƒ ( s ) μ1 ( ds ) = SÅ f ( s ) μ ( ds ) for E e 21. Let f be a μ1 - integrable function and let { f } be a sequence of -integrable simple functions ...
... function , it is evident that ƒ is also a u - integrable simple function , and that SÅ ƒ ( s ) μ1 ( ds ) = SÅ f ( s ) μ ( ds ) for E e 21. Let f be a μ1 - integrable function and let { f } be a sequence of -integrable simple functions ...
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