Linear Operators: General theory |
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Page 373
... Riesz [ 2 ; p . 456 ] . Lemma 4.2 is due to F. Riesz [ 8 ; p . 36 ] and was also used by Sz . - Nagy [ 5 ] . A similar argument may be applied to obtain this result in any uniformly convex B - space . This generalizes and abstracts a ...
... Riesz [ 2 ; p . 456 ] . Lemma 4.2 is due to F. Riesz [ 8 ; p . 36 ] and was also used by Sz . - Nagy [ 5 ] . A similar argument may be applied to obtain this result in any uniformly convex B - space . This generalizes and abstracts a ...
Page 388
... 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 and f → ƒ . This theorem remains valid in any uniformly convex B - space . Theorem 8.15 was ...
... 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 and f → ƒ . This theorem remains valid in any uniformly convex B - space . Theorem 8.15 was ...
Page 729
... Riesz [ 16 , 18 ] . Another proof , based on the interesting fact that a fixed point for a contraction in Hilbert space is also a fixed point for its adjoint , was given by Riesz and Sz . - Nagy [ 2 ] . Theorems for abstract vector ...
... Riesz [ 16 , 18 ] . Another proof , based on the interesting fact that a fixed point for a contraction in Hilbert space is also a fixed point for its adjoint , was given by Riesz and Sz . - Nagy [ 2 ] . Theorems for abstract vector ...
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 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