Math 6210: Solutions to Problem Set #3

Rudin, Chapter 4, Problem #3. The space L(T ) is separable since the trigonometric polynomials with complex coefficients whose real and imaginary parts are rational form a countable dense subset. (Denseness follows from Theorem 3.14 and Theorem 4.25; countability is clear since {e | n ∈ Z} is a countable basis of the trigonometric polynomials.) Meanwhile the space L(T ) is not separable. To see this, let S denote the set of indicator functions of subintervals of the circle (viewed as [0, 1)) with irrational endpoints. Then S is uncountable and the L distance between any two elements of S is 1. Any dense subset of L(T ) must contain an element in the (disjoint!) collection of balls {B1/3(x) |x ∈ S}. So any dense subset must be uncountable. ̃