Linear Operators: General theory |
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Page 373
... proved for closed linear manifolds in L2 [ 0 , 1 ] by E. Fischer [ 2 ] . The fact that a linear manifold which is not dense in the entire space has a non - zero orthogonal complement ( proved in 4.4 ) was proved without the assumption ...
... proved for closed linear manifolds in L2 [ 0 , 1 ] by E. Fischer [ 2 ] . The fact that a linear manifold which is not dense in the entire space has a non - zero orthogonal complement ( proved in 4.4 ) was proved without the assumption ...
Page 385
... proved independently by Cech [ 1 ] only slightly later . ( See also Stone [ 5 ] for an elementary treatment . ) Lemma 6.25 was proved in Stone [ 1 ; p . 465 ] -extensions of this re- sult are also found in Hewitt [ 5 ] and Kaplansky ...
... proved independently by Cech [ 1 ] only slightly later . ( See also Stone [ 5 ] for an elementary treatment . ) Lemma 6.25 was proved in Stone [ 1 ; p . 465 ] -extensions of this re- sult are also found in Hewitt [ 5 ] and Kaplansky ...
Page 462
... proved 3.9 after establishing Lemma 3.10 by induction . The case of Theorem 3.9 in which Y * and I ' 9 , was proved in the separa- ble case by Banach [ 1 ; p . 131 ] and in full generality by Alaoglu 1 ; p . 256 ] . Michael [ 1 ] gave ...
... proved 3.9 after establishing Lemma 3.10 by induction . The case of Theorem 3.9 in which Y * and I ' 9 , was proved in the separa- ble case by Banach [ 1 ; p . 131 ] and in full generality by Alaoglu 1 ; p . 256 ] . Michael [ 1 ] gave ...
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