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 x 。# y 。. We will prove that there exists an hoe H with h 。( x 。) ‡ 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 x 。# y 。. We will prove that there exists an hoe H with h 。( x 。) ‡ ho ( yo ) . If xo yo , then ...
Page 653
... Prove that limo t1T ( t ) x - x ] = y strongly , and thus ye D ( A ) . ∞ 3 Prove that 1 D ( 4 " ) is dense in S. ( Hint . Let X denote the class of functions K in C ( 0 , ∞ ) each vanishing outside a compact subset of ( 0 , ∞ ) . Prove ...
... Prove that limo t1T ( t ) x - x ] = y strongly , and thus ye D ( A ) . ∞ 3 Prove that 1 D ( 4 " ) is dense in S. ( Hint . Let X denote the class of functions K in C ( 0 , ∞ ) each vanishing outside a compact subset of ( 0 , ∞ ) . Prove ...
Page 670
... proves ( 3 ) . From ( 1 ) and ( 3 ) it is seen that ei + lg1 = g1 = e1 + 1 [ 1 − P ( gi + 1 + Pgi + 2 + ... ) ] ≥ 0 , gi 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 + lg1 = g1 = e1 + 1 [ 1 − P ( gi + 1 + Pgi + 2 + ... ) ] ≥ 0 , gi which proves ( i ) . To prove ( ii ) we shall first prove , by induction down- wards , that ( 4 ) ( gi + Pgi + 1 ...
Contents
A Settheoretic Preliminaries | 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 Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ