Introduction to Representation Theory
《Introduction to Representation Theory》 · 第 11–12 页 · Chapter 2，Section 2.3「Representations」，Proposition 2.3.9
Proposition 2.3.9 (Schur’s lemma). Let $V_1, V_2$ be representations of an algebra $A$ over any field $F$ (which need not be algebraically closed). Let $\phi\colon V_1\to V_2$ be a nonzero homomorphism of representations. Then: (i) If $V_1$ is irreducible, $\phi$ is injective. (ii) If $V_2$ is irreducible, $\phi$ is surjective. Thus, if both $V_1$ and $V_2$ are irreducible, $\phi$ is an isomorphism.