View source for Subrepresentation of a representation

A [[Linear representation|linear representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910201.png" /> in an invariant subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910202.png" /> of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910203.png" /> of a group (algebra, ring or semi-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910204.png" /> in a (topological) vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910205.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910206.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910208.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910209.png" /> is a [[Continuous representation|continuous representation]] (of a topological group, algebra, ring, or semi-group), then any subrepresentation of it is also continuous.