Linear Operators: General theory |
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Page 42
... proves the statement made above . We have seen that the mapping x → H ( x ) is a homomorphism . To show that it is an isomorphism , let a 。 yo . We will prove that there exists an hoe H with h 。( xo ) ‡ ho ( yo ) . If xo yo , then ...
... proves the statement made above . We have seen that the mapping x → H ( x ) is a homomorphism . To show that it is an isomorphism , let a 。 yo . We will prove that there exists an hoe H with h 。( xo ) ‡ ho ( yo ) . If xo yo , then ...
Page 653
... Prove that limo t1 [ T ( t ) x - x ] y strongly , and thus ye D ( A ) . 3 Prove that = ∞ n = 1 1 D ( 4 " ) is dense in S. ( Hint . Let X denote the class of functions K in C ( 0 , ∞ ) each vanishing outside a compact = { y y subset of ...
... Prove that limo t1 [ T ( t ) x - x ] y strongly , and thus ye D ( A ) . 3 Prove that = ∞ n = 1 1 D ( 4 " ) is dense in S. ( Hint . Let X denote the class of functions K in C ( 0 , ∞ ) each vanishing outside a compact = { y y subset of ...
Page 670
... proves ( 3 ) . From ( 1 ) and ( 3 ) it is seen that ei + 1gi = gi = ei +1 [ 1 −P ( gi + 1 + Pgi + 2 + . . . ) ] ≥ 0 , which proves ( i ) . To prove ( ii ) we shall first prove , by induction down- wards , that ( 4 ) ( gi + Pgi + 1 + ...
... proves ( 3 ) . From ( 1 ) and ( 3 ) it is seen that ei + 1gi = gi = ei +1 [ 1 −P ( gi + 1 + Pgi + 2 + . . . ) ] ≥ 0 , which proves ( i ) . To prove ( ii ) we shall first prove , by induction down- wards , that ( 4 ) ( gi + Pgi + 1 + ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ