Linear Operators: General theory |
From inside the book
Results 1-3 of 54
Page 388
... similar theorem valid in L ,, 1 < p < ∞o , which was proved by Radon [ 2 ; p . 1363 ] and F. Riesz [ 13 ] . THEOREM . If 1 < p < ∞ then a sequence { f } converges strongly to f in L , ( S , E , μ ) if and only if it converges weakly ...
... similar theorem valid in L ,, 1 < p < ∞o , which was proved by Radon [ 2 ; p . 1363 ] and F. Riesz [ 13 ] . THEOREM . If 1 < p < ∞ then a sequence { f } converges strongly to f in L , ( S , E , μ ) if and only if it converges weakly ...
Page 390
... similar results are due ( cf. Alexandroff [ 1 ; III ] ) . The following comments are pertinent to this theorem . Some remarks on X - convergence in X * . One of the problems set in Section 1 of this chapter is the determination of ...
... similar results are due ( cf. Alexandroff [ 1 ; III ] ) . The following comments are pertinent to this theorem . Some remarks on X - convergence in X * . One of the problems set in Section 1 of this chapter is the determination of ...
Page 539
... similar treat- ment for weakly compact operators is found in Bartle [ 2 ] . Operators with closed range . Some particular cases of these theo- rems were proved by Hellinger and Toeplitz [ 1 ] for l2 and by F. Riesz [ 2 , 6 ] for L , lp ...
... similar treat- ment for weakly compact operators is found in Bartle [ 2 ] . Operators with closed range . Some particular cases of these theo- rems were proved by Hellinger and Toeplitz [ 1 ] for l2 and by F. Riesz [ 2 , 6 ] for L , lp ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
80 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 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