Linear Operators: General theory |
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Page 104
It is customary to speak of the elements of F(S, E, (i, X) as if they were functions
rather than sets of equivalent functions and this we shall ordinarily do. Thus, we
shall write / instead of [/] and think of F(S, E, /i. X) as the set of all functions on S to
...
It is customary to speak of the elements of F(S, E, (i, X) as if they were functions
rather than sets of equivalent functions and this we shall ordinarily do. Thus, we
shall write / instead of [/] and think of F(S, E, /i. X) as the set of all functions on S to
...
Page 182
j£ f(s)p(ds) = j£ g(s)v(p, da) = ;.(£). Q.E.D. Note: The unique //-integrable function /
of Theorems 2 and 7 is called the Radon-Nikodym derivative of X with respect to
p and is often denoted by dkjdp. Thus dXjdp is defined /<-almost everywhere by ...
j£ f(s)p(ds) = j£ g(s)v(p, da) = ;.(£). Q.E.D. Note: The unique //-integrable function /
of Theorems 2 and 7 is called the Radon-Nikodym derivative of X with respect to
p and is often denoted by dkjdp. Thus dXjdp is defined /<-almost everywhere by ...
Page 196
For each s in S, F(s) is an equivalence class of functions, any pair of whose
members coincide A-almost everywhere. If for each s we select a particular
function f(s, □) € F(s), the resulting function f(s, t) defined on (R,Er,q) = (S,Es,fi)x(
T,ET,Z) ...
For each s in S, F(s) is an equivalence class of functions, any pair of whose
members coincide A-almost everywhere. If for each s we select a particular
function f(s, □) € F(s), the resulting function f(s, t) defined on (R,Er,q) = (S,Es,fi)x(
T,ET,Z) ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact
Bibliographic information
| Title | Linear Operators: General theory Volume 7 of Pure and applied mathematics Volume 1 of Linear Operators, Jacob T. Schwartz Volume 7 of Pure and applied mathematics Interscience Press |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of Michigan |
| Digitized | 20 Nov 2007 |
| Length | 2592 pages |
| Subjects | › › Mathematics / Algebra / Linear Mathematics / Calculus |
| Export Citation | BiBTeX EndNote RefMan |