Linear Operators: General theory |
From inside the book
Results 1-3 of 41
Page 293
... sequentially com- pact . It follows from Lemma II.3.27 that K is bounded . If the inte- grals Ef ( s ) μ ( ds ) are not countably additive uniformly with respect to ƒ in K there is a number ɛ > 0 , a decreasing sequence En € Σ with a ...
... sequentially com- pact . It follows from Lemma II.3.27 that K is bounded . If the inte- grals Ef ( s ) μ ( ds ) are not countably additive uniformly with respect to ƒ in K there is a number ɛ > 0 , a decreasing sequence En € Σ with a ...
Page 294
... sequentially compact then lim f ( s ) μ ( ds ) = 0 , μ ( E ) → 0 E uniformly for fin K. If μ ( S ) < ∞ then conversely this condition is suffi- cient for a bounded set K to be weakly sequentially compact . PROOF . Let K be a weakly ...
... sequentially compact then lim f ( s ) μ ( ds ) = 0 , μ ( E ) → 0 E uniformly for fin K. If μ ( S ) < ∞ then conversely this condition is suffi- cient for a bounded set K to be weakly sequentially compact . PROOF . Let K be a weakly ...
Page 314
... sequentially com- pact if and only if there exists a non - negative μ in ba ( S , E ) such that lim λ ( E ) = 0 μ ( E ) → 0 uniformly for λε Κ . PROOF . Let KC ba ( S , E ) be weakly sequentially compact and let V = UT be the isometric ...
... sequentially com- pact if and only if there exists a non - negative μ in ba ( S , E ) such that lim λ ( E ) = 0 μ ( E ) → 0 uniformly for λε Κ . PROOF . Let KC ba ( S , E ) be weakly sequentially compact and let V = UT be the isometric ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
31 other sections not shown
Other editions - View all
Common terms and phrases
A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ