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) | $.

References

[1]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , Acad. Press (1964) (Translated from Russian)