Linear Operators: General theory |
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Page 26
... identity mapping of X with the metric topology onto X with the given topology is continuous . Hence the identity is a homeomorphism and the space is metrizable . Q.E.D. 7. Convergence and Uniform Convergence of Generalized Sequences The ...
... identity mapping of X with the metric topology onto X with the given topology is continuous . Hence the identity is a homeomorphism and the space is metrizable . Q.E.D. 7. Convergence and Uniform Convergence of Generalized Sequences The ...
Page 249
... identity , follows immediately from the axioms . If d = inf x - k❘ the preceding identity shows that kЄK = k - k , 22x - k , | 2 + 2 | x ― k , | 2 - 4 \ x— ( k ̧ + k , ) / 2 | 2 | = ≤2x - k12 + 2x - k , 2—48 → 0 . Q.E.D. 3 DEFINITION ...
... identity , follows immediately from the axioms . If d = inf x - k❘ the preceding identity shows that kЄK = k - k , 22x - k , | 2 + 2 | x ― k , | 2 - 4 \ x— ( k ̧ + k , ) / 2 | 2 | = ≤2x - k12 + 2x - k , 2—48 → 0 . Q.E.D. 3 DEFINITION ...
Page 479
... identity on X * . Thus ( T * ) - 1 exists , is in B ( X * , ( T - 1 ) * . Conversely , if ( T * ) - 1 exists and is ... identity 2 2 ( Tx , y ) = ( x , T VI.2.6 479 ADJOINTS.
... identity on X * . Thus ( T * ) - 1 exists , is in B ( X * , ( T - 1 ) * . Conversely , if ( T * ) - 1 exists and is ... identity 2 2 ( Tx , y ) = ( x , T VI.2.6 479 ADJOINTS.
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