Linear Operators: General theory |
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Page 266
... preceding theorem and the dominated convergence theorem of Lebesgue ( III.6.16 ) . Q.E.D. 5 THEOREM . Let S be an arbitrary topological space and K a bounded subset of C ( S ) . Then K is conditionally compact if and only if for every ...
... preceding theorem and the dominated convergence theorem of Lebesgue ( III.6.16 ) . Q.E.D. 5 THEOREM . Let S be an arbitrary topological space and K a bounded subset of C ( S ) . Then K is conditionally compact if and only if for every ...
Page 668
... preceding theorem , it is assumed that p is measure preserving and metrically transitive then , for each † in L1 ( S , Σ , μ ) , the limit lim , A ( n ) f is almost everywhere equal to the constant space average Ssf ( s ) μ ( ds ) / μ ...
... preceding theorem , it is assumed that p is measure preserving and metrically transitive then , for each † in L1 ( S , Σ , μ ) , the limit lim , A ( n ) f is almost everywhere equal to the constant space average Ssf ( s ) μ ( ds ) / μ ...
Page 684
... preceding theorem satisfies the inequality m ( e ) ≤Ku ( e ) and thus the u - integrable function ƒ is also m - integrable . It was observed in the preceding proof that the limit g ( s ) exists for μ- με almost all s and hence it only ...
... preceding theorem satisfies the inequality m ( e ) ≤Ku ( e ) and thus the u - integrable function ƒ is also m - integrable . It was observed in the preceding proof that the limit g ( s ) exists for μ- με almost all s and hence it only ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ