Linear Operators: General theory |
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Page 360
... Suppose ( i ) , ( ii ) of the last exercise . Show that if ƒ is in BV then ( Sf ) ( x ) → f ( x ) at each point a where f is continuous . 25 Suppose that ( Sf ) ( x ) → ( 1/2 ) ( f ( x + ) + f ( x- ) ) for each step function f . Show ...
... Suppose ( i ) , ( ii ) of the last exercise . Show that if ƒ is in BV then ( Sf ) ( x ) → f ( x ) at each point a where f is continuous . 25 Suppose that ( Sf ) ( x ) → ( 1/2 ) ( f ( x + ) + f ( x- ) ) for each step function f . Show ...
Page 598
... Suppose that g , ( T ) converges in the uniform operator topology to a compact operator . Let λ € 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 Exercise ...
... Suppose that g , ( T ) converges in the uniform operator topology to a compact operator . Let λ € 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 Exercise ...
Page 718
... Suppose that Sh ( re ) Pd0 ≤K , 0 < r < 1 . Let h * ( 0 ) = max 。< r < 1 h ( re ) . Show that there exists an absolute constant C , such that f | h * ( 0 ) o d0 ≤ C ̧K . Show that lim , h ( re ) exists almost everywhere . Hint . Cf ...
... Suppose that Sh ( re ) Pd0 ≤K , 0 < r < 1 . Let h * ( 0 ) = max 。< r < 1 h ( re ) . Show that there exists an absolute constant C , such that f | h * ( 0 ) o d0 ≤ C ̧K . Show that lim , h ( re ) exists almost everywhere . Hint . Cf ...
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