Linear Operators: General theory |
From inside the book
Results 1-3 of 18
Page 289
... ( s ) μ ( ds ) , fe Lp . Applying Theorem 1 once more , this time to L * and L „ , we find there exists an he L , such that y * † = { sh ( s ) f ( s ) μ ( ds ) , ƒ € L ̧ · Thus x ** ( * ) = y * ( g ) = Ssg ( s ) h ( s ) μ ( ds ) = x * ( h ) ...
... ( s ) μ ( ds ) , fe Lp . Applying Theorem 1 once more , this time to L * and L „ , we find there exists an he L , such that y * † = { sh ( s ) f ( s ) μ ( ds ) , ƒ € L ̧ · Thus x ** ( * ) = y * ( g ) = Ssg ( s ) h ( s ) μ ( ds ) = x * ( h ) ...
Page 517
... ( s , t ) x * ( ds ) ] μ ( dt ) , for every Borel set E. 46 Let C - E C [ 0 , 1 ] and T € B ( C , C ) . Then there is ... ( Lp , C ) , B ( B ( S ) , C ) , B ( c , C ) , and B ( C , c ) where C L1 = L , [ 0 , 1 ] . C [ 0 , 1 ] and 48 ...
... ( s , t ) x * ( ds ) ] μ ( dt ) , for every Borel set E. 46 Let C - E C [ 0 , 1 ] and T € B ( C , C ) . Then there is ... ( Lp , C ) , B ( B ( S ) , C ) , B ( c , C ) , and B ( C , c ) where C L1 = L , [ 0 , 1 ] . C [ 0 , 1 ] and 48 ...
Page 722
... ( s ) exists almost everywhere for fe Lp , 1 ≤p < ∞ . Show that if S is a topological space , and if , for each λ > 0 , R ( 2 .; A ) f is continuous if f is continuous , then for 1 ≤ p < ∞ and ƒ € Lp the limit limo ( 2R ( 2 ; A ) f ) ( s ) ...
... ( s ) exists almost everywhere for fe Lp , 1 ≤p < ∞ . Show that if S is a topological space , and if , for each λ > 0 , R ( 2 .; A ) f is continuous if f is continuous , then for 1 ≤ p < ∞ and ƒ € Lp the limit limo ( 2R ( 2 ; A ) f ) ( s ) ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
59 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 closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ