Answer by Michael_1812 for Generalization of Schur's Lemma

In this lemma, the group G does not need to be finite.

The representation space V of an infinite group G may be either finite-dimensional or countable-dimensional (or, better to say, of dimension less than continuum). When the dimensionality is continuum, this lemma fails.

The proofs of these facts are surprisingly elementary, see the discussion at

Dixmier's lemma as a generalisation of Schur's first lemma