Solving linear function on nuo

Question

Problem:

Llinear functional on (N∪{0})

Let H = l²(N∪0)

(a) Show that if {a_n} ∈ H, then the power series Σ^∞_n=0  a_nzⁿ has radius of convergence > 1.

(b) If |λ| < 1 and  L: H→F (F is either the real or the complex field) is defined by L({a_n}) = Σ^∞_n=0 a_nλⁿ, find the vector h₀∈H such that L(h) = {h, h₀}∀ h ∈ H.

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

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