# Llinear functional on N U 0

Let H = l^2(N U 0)

(a) Show that if {a_n} is in H, then the power series sum_{n=0}^infty a_n z^n has radius of convergence >= 1.

(b) If |b| < 1 and linear functional L: H-->F (F is either the real or the complex field) is defined by L({a_n}) = sum_{n=0}^infty a_n b^n, find the vector h_0 in H such that L(h) = < h, h_0 > for all h in H.

(c) For a bounded linear functional L: H-->F define the norm of L as follows:

||L|| = sup {|L(h)|: for all h in H such that ||h||<1 }. What is the norm of the linear functional L defined in (b)?

https://brainmass.com/math/linear-algebra/llinear-functional-48823

#### Solution Preview

(a) The fact that H is l^2 means that the sum_0^infty |a_n|^2 converges; therefore |a_n|-->0 as n->0.

Therefore the power series sum_{n=0}^infty a_n z^n ...

#### Solution Summary

In this stepwise solution, the calculations are given for finding the vectors and norms of the linear function.