View source for Quasi-norm

A non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766001.png" /> defined on a [[Linear space|linear space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766002.png" /> and satisfying the same axioms as a [[Norm|norm]] except for the triangle inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766003.png" />, which is replaced by the weaker requirement: There exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766004.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766005.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766006.png" />.



====Comments====
The topology of a locally bounded topological vector space can be given by a quasi-norm. Conversely, a quasi-normed vector space is locally bounded. Here a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766007.png" /> in a topological vector space is bounded if for each open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766008.png" /> of zero there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766009.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660010.png" />, and a topological vector space is locally bounded if there is a bounded neighbourhood of zero. Given a circled bounded neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660011.png" /> of zero in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660012.png" /> (a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660013.png" /> is circled if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660015.png" />), the Minkowski functional of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660016.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660017.png" />. It is a quasi-norm.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,   "Topological vector spaces" , '''1''' , Springer  (1969)  pp. 159</TD></TR></table>