Fréchet space

Remarks

Equivalently, the topology of $E$ is generated by a countable family of seminorms $p_1 \le p_2 \le \cdots$ that separates points, and $E$ is complete. From such seminorms one gets a translation-invariant metric $$d(x, y) = \sum_{n = 1}^{\infty} 2^{-n} \frac{p_n(x - y)}{1 + p_n(x - y)}$$ that induces the topology. Example: the space $C(\R)$ of continuous real functions with the seminorms $p_n(f) = \sup_{\abs{t} \le n} \abs{f(t)}$ is a Fréchet space. A sequence that is Cauchy for all $p_n$ converges uniformly on each $[-n, n]$, and the limit is continuous. Every Banach space is a Fréchet space. Non-example: $L^p([0, 1])$ with $0 < p < 1$ is an F-space but is not locally convex, so it is not a Fréchet space. Some older texts use the name Fréchet space for what is here called an F-space.