Linear Operators: General theory |
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Page 254
... seen that v is equivalent to vg and thus that υβ v is in V. Since { v } is a basis , the vector u , has an expansion of the form u ( u , v ) , so that u , is in the closed linear manifold de- termined by those vg with ( u , v ) 0. Since ...
... seen that v is equivalent to vg and thus that υβ v is in V. Since { v } is a basis , the vector u , has an expansion of the form u ( u , v ) , so that u , is in the closed linear manifold de- termined by those vg with ( u , v ) 0. Since ...
Page 286
... seen from the Vitali - Hahn- Saks theorem ( III.7.2 ) that lim g ( s ) f ( s ) μu ( ds ) = 0 μ ( E ) -0 E uniformly in n . Since it is assumed for the present that μ ( S ) < ∞ it is seen from Theorem III.6.15 that fg is in L1 and that ...
... seen from the Vitali - Hahn- Saks theorem ( III.7.2 ) that lim g ( s ) f ( s ) μu ( ds ) = 0 μ ( E ) -0 E uniformly in n . Since it is assumed for the present that μ ( S ) < ∞ it is seen from Theorem III.6.15 that fg is in L1 and that ...
Page 715
... seen from Theorem 5 that all the points of σ ( T * ) of unit modulus are roots of unity , and by Lemma VII.3.7 the same holds for σ ( T ) . The remaining conclusions in Theorems 6 and 7 are . then provided exactly as before , since no ...
... seen from Theorem 5 that all the points of σ ( T * ) of unit modulus are roots of unity , and by Lemma VII.3.7 the same holds for σ ( T ) . The remaining conclusions in Theorems 6 and 7 are . then provided exactly as before , since no ...
Contents
Preliminary Concepts | 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 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