View source for Countably-normed space

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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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Gel'fand,   G.E. Shilov,   "Generalized functions" , Acad. Press  (1964)  (Translated from Russian)</TD></TR></table>