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A [[Linear representation|linear representation]]  $  \rho $
in an invariant subspace  $  F \subset  E $
of a representation  $  \pi $
of a group (algebra, ring or semi-group)  $  X $
in a (topological) vector space  $  E $
defined by the formula  $  \rho ( x) \xi = \pi ( x) \xi $
for all  $  \xi \in F $, 
$  x \in X $. 
If  $  \pi $
is a [[Continuous representation|continuous representation]] (of a topological group, algebra, ring, or semi-group), then any subrepresentation of it is also continuous.