Linear representation

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A homomorphism $\pi$ of a group (respectively an algebra, ring, semi-group) $X$ into the group of all invertible linear operators on a vector space $E$ (respectively, into the algebra, ring, multiplicative semi-group of all linear operators on $E$). If $E$ is a topological vector space, then a linear representation of $X$ on $E$ is a representation whose image contains only continuous linear operators on $E$. The space $E$ is called the representation space of $\pi$ and the operators $\pi(x)$, $x\in X$, are called the operators of the representation $\pi$.


[1] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)



[a1] C.W. Curtis, I. Reiner, "Methods of representation theory" , 1–2 , Wiley (Interscience) (1981–1987)
[a2] J.-P. Serre, "Répresentations linéaires des groupes finis" , Hermann (1967)
[a3] A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian)
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Linear representation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article