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Page 464
... satisfies inf Rf ( x ) ≤ RP ( f ) ≤ sup Rf ( x ) , xЄK also satisfies Þ ( f ) = f ( xo ) , XEK fer je T for some xe K. ( 4 ) Iƒ K¿ , § < 0 , 0 a limit ordinal , is a monotone decreasing trans- finite sequence of T - closed convex sets ...
... satisfies inf Rf ( x ) ≤ RP ( f ) ≤ sup Rf ( x ) , xЄK also satisfies Þ ( f ) = f ( xo ) , XEK fer je T for some xe K. ( 4 ) Iƒ K¿ , § < 0 , 0 a limit ordinal , is a monotone decreasing trans- finite sequence of T - closed convex sets ...
Page 495
... satisfies properties ( i ) and ( ii ) above so that B ( h ) = Bo . This proves that if he C ( S ) and ƒ e Bo , then fh e Bo . Now let ƒ be a fixed element of Bo , and let B ( f ) = { g € Bolfg € Bo } . We have just proved that C ( S ) ...
... satisfies properties ( i ) and ( ii ) above so that B ( h ) = Bo . This proves that if he C ( S ) and ƒ e Bo , then fh e Bo . Now let ƒ be a fixed element of Bo , and let B ( f ) = { g € Bolfg € Bo } . We have just proved that C ( S ) ...
Page 557
... satisfies R1 ( T ) = 0. In the same way , the product R2 of all the factors ( 2-2 ) , where B min ( x , v ( 2 ) ) , satisfies R ( T ) = 0. Since any polynomial P having a zero of order v ( 2 ) at each λ € σ ( T ) is divisible by R2 ...
... satisfies R1 ( T ) = 0. In the same way , the product R2 of all the factors ( 2-2 ) , where B min ( x , v ( 2 ) ) , satisfies R ( T ) = 0. Since any polynomial P having a zero of order v ( 2 ) at each λ € σ ( T ) is divisible by R2 ...
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