View source for Countably-normed space

A locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267401.png" /> whose topology is defined using a countable set of compatible norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267402.png" /> i.e. norms such that if a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267403.png" /> that is fundamental in the norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267405.png" /> converges to zero in one of these norms, then it also converges to zero in the other. The sequence of norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267406.png" /> can be replaced by a non-decreasing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267407.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267408.png" />, which generates the same topology with base of neighbourhoods of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267409.png" />. A countably-normed space is metrizable, and its metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c02674010.png" /> can be defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c02674011.png" /></td> </tr></table>

An example of a countably-normed space is the space of entire functions that are analytic in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c02674012.png" /> with the topology of uniform convergence on any closed subset of this disc and with the collection of norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c02674013.png" />.

====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>