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Let {u₁, 2,...} be an orthonormal set in a Hilbert space H and (kn) be a sequence in K. If En|kn| <∞o, then by 8.1, the series En knun converges in H. But the condition En kn<∞ is not necessary for the convergence of the series knun in H. In fact, as we have seen above, Σ kn2<∞o is a necessary and sufficient condition for such a convergence. For example, if kn = 1/n for n = 1,2,..., then En knl is divergent, but n knun ccnverges in H, since Στη || <00. Request explain how.

Let {u₁, 2,...} be an orthonormal set in a Hilbert space H and (kn) be a sequence in K. If En|kn| <∞o, then by 8.1, the series En knun converges in H. But the condition En kn<∞ is not necessary for the convergence of the series knun in H. In fact, as we have seen above, Σ kn2<∞o is a necessary and sufficient condition for such a convergence. For example, if kn = 1/n for n = 1,2,..., then En knl is divergent, but n knun ccnverges in H, since Στη || <00. Request explain how.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Request explain the underlined portion from the book "Functional Analysis" by BV Limaye

 

Let {u₁, ₂,...} be an orthonormal set in a Hilbert space H and (kn) be a sequence in K. If Σ kn<∞o, then by 8.1, the series En knun converges in H. But the condition Σ kn<∞ is not necessary for the convergence of the series knun in H. In fact, as we have seen above, Σn kn2<∞o is a necessary and sufficient condition for such a convergence. For example, if kn = 1/n for n = 1,2,..., then En knl is divergent, but knun ccnverges in H, since Σn |kn|² <∞0. Request explain how.
Transcribed Image Text:Let {u₁, ₂,...} be an orthonormal set in a Hilbert space H and (kn) be a sequence in K. If Σ kn<∞o, then by 8.1, the series En knun converges in H. But the condition Σ kn<∞ is not necessary for the convergence of the series knun in H. In fact, as we have seen above, Σn kn2<∞o is a necessary and sufficient condition for such a convergence. For example, if kn = 1/n for n = 1,2,..., then En knl is divergent, but knun ccnverges in H, since Σn |kn|² <∞0. Request explain how.
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