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 seriesEn knun converges in H. But the condition Σ kn<∞ is notnecessary for the convergence of the series knun in H. In fact,as we have seen above, Σn kn2<∞o is a necessary and sufficientcondition for such a convergence. For example, if kn = 1/n for n =1,2,..., then En knl is divergent, but knun ccnverges in H, sinceΣn |kn|² <∞0.Request explain how.

The given question is related with functional analysis. Given that kn = 1n for n = 1, 2, … , . We…

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.