Linear Operators: General theory |
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Page 421
... Hence .... Yn ] = Σai Yi • i = 1 Q.E.D. - n g ( x ) = Σα2t , ( x ) , x € X. PROOF OF THEOREM 9 Every functional in ... hence xo Є g ( x ) 1. Since is a linear space , it contains ma , for every integer m . Hence m g ( x ) = g ( mx ) < 1 ...
... Hence .... Yn ] = Σai Yi • i = 1 Q.E.D. - n g ( x ) = Σα2t , ( x ) , x € X. PROOF OF THEOREM 9 Every functional in ... hence xo Є g ( x ) 1. Since is a linear space , it contains ma , for every integer m . Hence m g ( x ) = g ( mx ) < 1 ...
Page 441
... hence a compact , subset of co ( Q ) . Hence K = i co ( Q ) = co ( K1U ... UK1 ) = co ( K1 ○ ... ... U K „ ) , by an easy induction on Lemma 2.5 . It follows readily that p has the form p = Στ Σak ,, a ; ≥0 , Σa , = 1 , k ̧ e K ; and ...
... hence a compact , subset of co ( Q ) . Hence K = i co ( Q ) = co ( K1U ... UK1 ) = co ( K1 ○ ... ... U K „ ) , by an easy induction on Lemma 2.5 . It follows readily that p has the form p = Στ Σak ,, a ; ≥0 , Σa , = 1 , k ̧ e K ; and ...
Page 485
... Hence y Tx ( T * y * ) x converges uniformly for a in S. It follows that T * y * , converges in the strong topology of X * . Thus T * is compact . * - Conversely , let T * be compact . Then , by the point just proved , T ** is compact ; ...
... Hence y Tx ( T * y * ) x converges uniformly for a in S. It follows that T * y * , converges in the strong topology of X * . Thus T * is compact . * - Conversely , let T * be compact . Then , by the point just proved , T ** is compact ; ...
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