(a) A separable Banach algebra.

(b) A non-separable Banach algebra.

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(a) {r_{1} + ir_{2}; r_{1}, r_{2} ∈ Q} is a countable dense subset of C. Hence C is a separable Banach algebra.

(b) l^{∞} isn’t separable. In fact if S = {a_{1}, a_{2}, ⋯, } is a countable set in

l^{∞}, a_{n} = (a_{n}^{k})_{k∈N}, and b_{n} = \{ \begin{matrix}0 & |a_{n}^{n}| ≥ 1 \\ 2 & |a_{n}^{n}| < 1 \end{matrix} , then b = (b_{n}) ∈ l^{∞} and for all

n, ∥ b − a_{n} ∥_{∞}≥ |b_{n} − a_{n}^{n}| ≥ 1. So that the neighborhood of b with the radius 1 doesn’t intersect S. Thus S isn’t dense in l^{∞}. For another proof see [A&B, Problem 25.7].

Ref. [A&B] C.D. Aliprantis and O. Burkinshaw, Problems in real analysis, Acad. Press, 1990.