Linear Operators: General theory |
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Page 171
... Suppose that every continuous function is u - measurable . Show that * contains every Borel set . 24 Suppose that S is a compact topological space , with the property that for every covering of S by a finite number of open sets G1 ...
... Suppose that every continuous function is u - measurable . Show that * contains every Borel set . 24 Suppose that S is a compact topological space , with the property that for every covering of S by a finite number of open sets G1 ...
Page 360
... Suppose that ( i ) ( Snf ) ( x ) → f ( x ) uniformly in a for f in AC . ( ii ) The convergence of Sf is localized . Show that for any f in CBV , ( S „ f ) ( x ) → f ( x ) uniformly in æ . 24 Suppose ( i ) , ( ii ) of the last exercise ...
... Suppose that ( i ) ( Snf ) ( x ) → f ( x ) uniformly in a for f in AC . ( ii ) The convergence of Sf is localized . Show that for any f in CBV , ( S „ f ) ( x ) → f ( x ) uniformly in æ . 24 Suppose ( i ) , ( ii ) of the last exercise ...
Page 598
... Suppose that gn ( T ) converges in the uniform operator topology to a compact operator . Let 2 e σ ( T ) , and lim → ( 2 ) 0. Show that is a pole of σ ( T ) , and that E ( 2 ; T ) X has a positive finite dimension . ( Hint . See ...
... Suppose that gn ( T ) converges in the uniform operator topology to a compact operator . Let 2 e σ ( T ) , and lim → ( 2 ) 0. Show that is a pole of σ ( T ) , and that E ( 2 ; T ) X has a positive finite dimension . ( Hint . See ...
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