Every unitary representation is a direct sum of cyclic representations.

Question

Every unitary representation is a direct sum of cyclic representations. it can be proved without the Zorn's Lemma ?

Answer 1

I don't think you need Zorn's Lemma. Let $V$ be a unitary representation of the group $G$ and let $U$ be the subrepresentation that is the direct sum of all cyclic subrepresentations. Suppose for contradiction that $U \not = V$. Then, $$V = U \oplus U^{\perp}$$ with $U^{\perp}$ nonzero. So, pick a vector $v \in U^{\perp}$. Then, $\text{span}\{G\cdot v\}$ is a cyclic subrepresentation of $V$ contained in $U^{\perp}$ and hence not contained in $U$, a contradiction. Hence, $U = V$ and hence every unitary representation is a direct sum of its cyclic subrepresentations.

Actually, note that in the above argument, unitary isn't necessary. All you need is complete reducibility i.e. every subrepresentation of $V$ has a complementary subrepresentation.