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Page 103
space of all functions which map S into 3 (see I.6.1). Unfortunately, this is rarely
the case and so a slight detour will be made. 3 DEFINITION. The function f on S
to 3 is said to be a u-null function or, when u is understood, simply a null function
if ...
space of all functions which map S into 3 (see I.6.1). Unfortunately, this is rarely
the case and so a slight detour will be made. 3 DEFINITION. The function f on S
to 3 is said to be a u-null function or, when u is understood, simply a null function
if ...
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, -) e F(s), the resulting function f(s, t) defined on (R, 2r, 9) = (S, As. Au)
× (T, 21, ...
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, -) e F(s), the resulting function f(s, t) defined on (R, 2r, 9) = (S, As. Au)
× (T, 21, ...
Page 199
integral ss f(s,t)a(ds), as a function of t, is equal to the element Js F(s) u(ds) of L,(T
. 2 r. 2, 3). PRoof. Let Xr be partitioned into a sequence {E,} of disjoint sets of finite
Z-measure. For 1 < p < 20 let L, H Lp(T. 2 r, 2, 3) and define the maps U, n = 1, ...
integral ss f(s,t)a(ds), as a function of t, is equal to the element Js F(s) u(ds) of L,(T
. 2 r. 2, 3). PRoof. Let Xr be partitioned into a sequence {E,} of disjoint sets of finite
Z-measure. For 1 < p < 20 let L, H Lp(T. 2 r, 2, 3) and define the maps U, n = 1, ...
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Contents
A Settheoretic Preliminaries | 1 |
Convergence and Uniform Convergence of Generalized | 26 |
Algebraic Preliminaries | 34 |
Copyright | |
27 other sections not shown
Common terms and phrases
additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero
Bibliographic information
| Title | Linear Operators, Part 1 Linear Operators, Nelson Dunford Volume 1 of Linear Operators: General Theory. Spectral Theory, Self Adjoint Operators in Hilbert Space. Spectral Operators. I. II. III, Jacob T. Schwartz Volume 7 of Pure and applied mathematics : a series of texts and monographs Pure and applied mathematics, ISSN 0079-8185 Wiley classics library |
| Authors | Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle |
| Edition | reprint |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 26 Jun 2018 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |