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    Llinear functional on N U 0

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    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)?

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    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.