# Countably-normed space

A locally convex space $X$ whose topology is defined using a countable set of compatible norms $\| * \| _ {1} \dots \| * \| _ {n} \dots$ i.e. norms such that if a sequence $\{ x _ {n} \} \subset X$ that is fundamental in the norms $\| * \| _ {p}$ and $\| * \| _ {q}$ converges to zero in one of these norms, then it also converges to zero in the other. The sequence of norms $\{ \| * \| _ {n} \}$ can be replaced by a non-decreasing sequence $\| * \| _ {p} \leq \| * \| _ {q}$, where $p < q$, which generates the same topology with base of neighbourhoods of zero $U _ {p, \epsilon } = \{ {x \in X } : {\| x \| _ {p} < \epsilon } \}$. A countably-normed space is metrizable, and its metric $\rho$ can be defined by
$$\rho ( x, y) = \ \sum _ {n = 1 } ^ \infty { \frac{1}{2 ^ {n} } } \frac{\| x - y \| _ {n} }{1 + \| x - y \| _ {n} } .$$
An example of a countably-normed space is the space of entire functions that are analytic in the unit disc $| z | < 1$ with the topology of uniform convergence on any closed subset of this disc and with the collection of norms $\| x ( z) \| _ {n} = \max _ {| z | \leq 1 - 1 / n } | x ( z) |$.