Linear Operators: General theory |
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Page 73
... s space Show that if the norm of s be defined as = [ $ 1 , 82 , . . . ] . m is a B - space . [ s ] = lub s , 1 < i ... LIM Sn ( a ) LIM s - n∞ LIM Sn + 1 ; N1xx ∞4126 = * ( s ) , show that ( b ) LIM ( as , + ẞtn ) = a LIM sn + ẞ LIM ...
... s space Show that if the norm of s be defined as = [ $ 1 , 82 , . . . ] . m is a B - space . [ s ] = lub s , 1 < i ... LIM Sn ( a ) LIM s - n∞ LIM Sn + 1 ; N1xx ∞4126 = * ( s ) , show that ( b ) LIM ( as , + ẞtn ) = a LIM sn + ẞ LIM ...
Page 339
... lim , ( ) , 1 , 2 , ... all exist , and that such a sequence con- verges weakly to the element x = { } . Show that if p 1 , the same condition describes co - convergence in c . = = 5 Show that no space B ( S ... ( S ) is weakly complete unless ...
... lim , ( ) , 1 , 2 , ... all exist , and that such a sequence con- verges weakly to the element x = { } . Show that if p 1 , the same condition describes co - convergence in c . = = 5 Show that no space B ( S ... ( S ) is weakly complete unless ...
Page 352
Nelson Dunford, Jacob T. Schwartz. ( d ) lim fo K ( x , y ) dy exists . 818 Show that foK ( x , y ) f ( y ) dy has the same limit at a = ∞ as f ( x ) if and only if we have and ( b ′ ) lim s ^ K ( x , y ) dy = 0 for all 0 < A < ∞ ∞4x ( d ...
Nelson Dunford, Jacob T. Schwartz. ( d ) lim fo K ( x , y ) dy exists . 818 Show that foK ( x , y ) f ( y ) dy has the same limit at a = ∞ as f ( x ) if and only if we have and ( b ′ ) lim s ^ K ( x , y ) dy = 0 for all 0 < A < ∞ ∞4x ( d ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ