Linear 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)?

Proof that a sequence is monotone increasing

(See attached file for full problem description and equations) --- Prove that the sequence is monotone increasing. Use the following hints: 1) If ln f(x) is increasing, then so is f(x). 2) If , then f is increasing. 3) ln x is defined to be . ---

Area measure

Let m be an area measure on {z in C:|z| < 1}. Show that 1, z, z^2,... are orthogonal vectors in L^2(m). Find ||z^n||, n >= 0. If e_n=(z^n)/||z^n||, n >= 0, is {e_0, e_1,...} a basis for L^2(m)?

Convergence of Sequence

(See attached file for full problem description and equations) --- Prove that the sequence of functions … converges for every , and find the limit to which it converges. Please show each step! Thank you ---

Proof of Uniform Convergence of a Sequence

(See attached file for full problem description and equations) --- Let f be a function such that on and let for . Prove that uniformly on . Please show each step! Thanks ---