1. Let X be a Hilbert space and let {n n = : ∞ 1,2...} be an orthogonal subset of X. N Show that the series ✓ ✓ is convergent in X, that is lim Σ exists, if and only if Σ ||xk||² < ∞. k=1 k=1 ∞0+N k=1

Find all monic irreducible polynomials of degree 2 over Z3.

A series ∞ n=0 an is said to be Abel-summable to L if the power series converges for all x ∈ [0, 1) and L = limx→1− f(x).

3 a) Suppose (an) is Cauchy and that for every k ∈ N, the interval (−1/k, 1/k) contains at least one term of (an). Can we say that (an) converges to 0? Either show that it does or give a counter-example.