Linear Operators: General theory |
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Page 421
... Hence 191 ... 9 Yn ] Σai Yi i = 1 Q.E.D. n g ( x ) = Σx ; f ; ( x ) , i = 1 x € X. PROOF OF THEOREM 9. Every ... hence g ( ) < 1. Since 1 , is a linear space , it contains ma 。 for every integer m . Hence m g ( x ) = g ( mao ) 1 , from ...
... Hence 191 ... 9 Yn ] Σai Yi i = 1 Q.E.D. n g ( x ) = Σx ; f ; ( x ) , i = 1 x € X. PROOF OF THEOREM 9. Every ... hence g ( ) < 1. Since 1 , is a linear space , it contains ma 。 for every integer m . Hence m g ( x ) = g ( mao ) 1 , from ...
Page 423
... Hence y ( Tx ) < e , so that Tæ e N ( 0 ; y * , . . . , y * , ε ) . Therefore , Tis weakly continuous at the origin , and hence at every point . Conversely , suppose that T is weakly continuous , and y * € Y * . Then y * T is a linear ...
... Hence y ( Tx ) < e , so that Tæ e N ( 0 ; y * , . . . , y * , ε ) . Therefore , Tis weakly continuous at the origin , and hence at every point . Conversely , suppose that T is weakly continuous , and y * € Y * . Then y * T is a linear ...
Page 441
... hence a compact , subset of co ( Q ) . Hence = co ( Q ) = co ( K1 U ...... .U K „ ) ... - = co ( K1U ... UK „ ) , by an easy induction on Lemma 2.5 . It follows readily that p has the form p Σ = 1 a¡k1 , a¡ ≥ 0 , Σ " = 1a¿ 1 , k , € K1 ...
... hence a compact , subset of co ( Q ) . Hence = co ( Q ) = co ( K1 U ...... .U K „ ) ... - = co ( K1U ... UK „ ) , by an easy induction on Lemma 2.5 . It follows readily that p has the form p Σ = 1 a¡k1 , a¡ ≥ 0 , Σ " = 1a¿ 1 , k , € K1 ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ