Linear Operators: General theory |
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Page 127
... unbounded if sup u ( EE ) + ; otherwise call it bounded . Then either ΕΕΣ = ( a ) every unbounded set contains an unbounded set of arbitrarily large measure , or ( b ) there exists an unbounded set FeΣ and an integer N such that F ...
... unbounded if sup u ( EE ) + ; otherwise call it bounded . Then either ΕΕΣ = ( a ) every unbounded set contains an unbounded set of arbitrarily large measure , or ( b ) there exists an unbounded set FeΣ and an integer N such that F ...
Page 601
... unbounded . The last statement is evident from the definition of Ø . Q.E.D. 3 DEFINITION . For fe F ( T ) we define f ( T ) = q ( 4 ) , where 9 € F ( A ) is given by q ( u ) = f ( Þ − 1 ( μ ) ) . 4 THEOREM . If f is in F ( T ) then f ...
... unbounded . The last statement is evident from the definition of Ø . Q.E.D. 3 DEFINITION . For fe F ( T ) we define f ( T ) = q ( 4 ) , where 9 € F ( A ) is given by q ( u ) = f ( Þ − 1 ( μ ) ) . 4 THEOREM . If f is in F ( T ) then f ...
Page 612
... unbounded self adjoint operators ; these results were com- pleted by Heinz [ 1 ] . See also Newburgh [ 1 , 2 ] . Most of these results pertain to the point spectrum . The con- tinuous spectrum is much more difficult to handle , but has ...
... unbounded self adjoint operators ; these results were com- pleted by Heinz [ 1 ] . See also Newburgh [ 1 , 2 ] . Most of these results pertain to the point spectrum . The con- tinuous spectrum is much more difficult to handle , but has ...
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