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Page 193
12 Lemma. Let (R, ZR, q) be the product of f inite measure spaces (S, Z, yu) and (
T, ZT, X). Let E be a g-null set in R. Then for X-almost all t, the set E(t) = {s\[s, t] e E
) is a fi-null set. Proof. By Lemma 11 it may be assumed that the measure ...
12 Lemma. Let (R, ZR, q) be the product of f inite measure spaces (S, Z, yu) and (
T, ZT, X). Let E be a g-null set in R. Then for X-almost all t, the set E(t) = {s\[s, t] e E
) is a fi-null set. Proof. By Lemma 11 it may be assumed that the measure ...
Page 699
1 f° _ 2j_ , 1 e~v do = - \/n 2 Therefore Vi □ = ar»/*<; \*y/m VM) fa 2y2' 2V^Jo ^/SI
2-v/: 2<r-V« /•<» ^ _/Vl*_\« — e V«« I dy Jo 2y9 V71 Jo A = «-V«, which proves (*)
and completes the proof of the lemma. Q.E.D. We shall now state and prove the ...
1 f° _ 2j_ , 1 e~v do = - \/n 2 Therefore Vi □ = ar»/*<; \*y/m VM) fa 2y2' 2V^Jo ^/SI
2-v/: 2<r-V« /•<» ^ _/Vl*_\« — e V«« I dy Jo 2y9 V71 Jo A = «-V«, which proves (*)
and completes the proof of the lemma. Q.E.D. We shall now state and prove the ...
Page 700
amJ0 Jo a* Jo Jo This shows that if the lemma is known to be true for an integer ft
it is also true for any integer m ^ ft. Thus, to prove the lemma, it will suffice to show
that it is true if k is even. If ft = 1 the lemma has already been proved (Lemma ...
amJ0 Jo a* Jo Jo This shows that if the lemma is known to be true for an integer ft
it is also true for any integer m ^ ft. Thus, to prove the lemma, it will suffice to show
that it is true if k is even. If ft = 1 the lemma has already been proved (Lemma ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations disjoint Doklady Akad Duke Math element ergodic theorem exists finite dimensional function defined Hausdorff space Hence Hilbert space homeomorphism ibid inequality integral Lebesgue Lebesgue measure Lemma linear functional linear map linear operator linear topological space measurable function 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 Sbornik N. S. scalar self-adjoint semi-group sequentially compact Show simple functions spectral Studia Math subset subspace Suppose theory topological space Trans uniformly Univ valued function Vber vector space weak topology weakly compact zero
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 California |
| Digitized | 16 Mar 2011 |
| Length | 2592 pages |
| Subjects | › › Algebra, Universal Algebras, Linear Linear operators Mathematical analysis Mathematics / Algebra / Linear Mathematics / Mathematical Analysis |
| Export Citation | BiBTeX EndNote RefMan |