Fréchet algebra

Remarks

The algebra is called locally multiplicatively convex (or m-convex) if the topology is given by seminorms with $p_n(ab) \le p_n(a) p_n(b)$. Some authors include this condition in the definition. Every Banach algebra is a Fréchet algebra, with the single seminorm given by the norm.

Example: $A = C(\R)$, the continuous complex functions on $\R$, with $p_n(f) = \max_{\abs{x} \le n} \abs{f(x)}$. Each $p_n$ is a seminorm with $p_n(fg) \le p_n(f) p_n(g)$, and the sequence is increasing. Convergence for all $p_n$ is uniform convergence on compact sets, and a locally uniform limit of continuous functions is continuous, so $A$ is complete. So $C(\R)$ is an m-convex Fréchet algebra.

Non-example: the polynomials $\C[x]$ with the same seminorms satisfy every condition except completeness in clause 1. The Taylor polynomials of $e^x$ form a Cauchy sequence for every $p_n$, but their limit $e^x$ is not a polynomial.