An invariant subspace of a representation is a vector subspace satisfying for every represented group element .
A nonzero representation is reducible when it has a nonzero proper invariant subspace; otherwise it is an irreducible representation.
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In functional analysis and operator theory, an **invariant subspace** refers to a subspace of a given vector space that is preserved under the action of a given linear operator. More formally, let \( T: V \to V \) be a linear operator on a vector space \( V \).