Linear Operators: General theory |
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Page 89
... seen to be a topological linear space . If X and Y are B- ( or F- ) spaces , then XY is a B- ( or an F- ) space under either of the norms | [ x , y ] | = max ( | x | , | y | ) , [ [ x , y ] ] = { xP + y } , 1 ≤ p < ∞ , and these norms ...
... seen to be a topological linear space . If X and Y are B- ( or F- ) spaces , then XY is a B- ( or an F- ) space under either of the norms | [ x , y ] | = max ( | x | , | y | ) , [ [ x , y ] ] = { xP + y } , 1 ≤ p < ∞ , and these norms ...
Page 254
... seen that vg is equivalent to v and thus that V is in V. Since { v } is a basis , the vector u has an expansion of the form u ( u , v ) Vg , so that u , is in the closed linear manifold de- termined by those up with ( u , v ) u̟ € sp ...
... seen that vg is equivalent to v and thus that V is in V. Since { v } is a basis , the vector u has an expansion of the form u ( u , v ) Vg , so that u , is in the closed linear manifold de- termined by those up with ( u , v ) u̟ € sp ...
Page 567
... seen in the proof of Lemma 2 that if | μ | < | R ( 2 ; T ) | −1 , then λ + μ € o ( T ) . Hence d ( λ ) ≥ | R ( 2 ; T ) | -1 , from which the statements follow . Q.E.D. n - x∞ 4 LEMMA . The closed set o ( T ) is bounded and non - void ...
... seen in the proof of Lemma 2 that if | μ | < | R ( 2 ; T ) | −1 , then λ + μ € o ( T ) . Hence d ( λ ) ≥ | R ( 2 ; T ) | -1 , from which the statements follow . Q.E.D. n - x∞ 4 LEMMA . The closed set o ( T ) is bounded and non - void ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ