[proofplan] Let $V$ be an arbitrary finite-dimensional continuous irreducible representation of $G = \operatorname{SU}(2)$. Restricting to the maximal torus $T \cong S^1$ and decomposing into $T$-weight spaces shows that the character $\chi_V$ (restricted to $T$) is an even Laurent polynomial in $z \in S^1$ with non-negative integer coefficients — i.e., $\chi_V \in \mathbb{N}[z, z^{-1}]_{\mathrm{ev}}$. The known characters $\chi_n$ of $V_n$, namely $\chi_n(z) = z^n + z^{n-2} + \cdots + z^{-n}$, form a $\mathbb{Q}$-basis of $\mathbb{Q}[z, z^{-1}]_{\mathrm{ev}}$ because the leading-term filtration $\chi_n \mapsto z^n$ is an upper-triangular bijection. So $\chi_V = \sum_n a_n \chi_n$ with rational $a_n$. Clearing denominators and rearranging into an equality of characters of representations, then invoking unique decomposition into irreducibles (Maschke for compact groups together with the fact that characters separate isomorphism classes), forces $V \cong V_n$ for a single $n$. [/proofplan]

[step:Restrict $V$ to the maximal torus and identify $\chi_V$ with an even Laurent polynomial]Let $V$ be a finite-dimensional continuous irreducible representation of $G = \operatorname{SU}(2)$, and let $\chi_V: G \to \mathbb{C}$ be its character. The character $\chi_V$ is a class function on $G$, so by the [Conjugacy in $\operatorname{SU}(2)$](/theorems/2475) classification, $\chi_V$ is determined by its restriction to the maximal torus \begin{align*} T = \left\{ t_z := \mathrm{diag}(z, z^{-1}) : z \in S^1 \right\} \subseteq G. \end{align*} Restricting $V$ to $T$ gives a continuous representation $V|_T$. Since $T \cong S^1$ is a compact abelian Lie group, $V|_T$ decomposes as a direct sum of one-dimensional $T$-representations (by the character theory of $S^1$, every continuous representation of $S^1$ is a direct sum of characters $z \mapsto z^k$ with $k \in \mathbb{Z}$). Write \begin{align*} V|_T \cong \bigoplus_{k \in \mathbb{Z}} \mathbb{C}^{m_k}, \quad m_k \in \mathbb{Z}_{\geq 0}, \end{align*} where $\mathbb{C}^{m_k}$ is the $k$-eigenspace, with $T$ acting on each summand via $t_z \mapsto z^k$. The character of this $T$-representation is \begin{align*} \chi_V(t_z) = \sum_{k \in \mathbb{Z}} m_k z^k. \end{align*} Only finitely many $m_k$ are nonzero (since $V$ is finite-dimensional), so this is a Laurent polynomial in $z$. *Evenness.* The element $s = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \in G$ satisfies $s t_z s^{-1} = t_{z^{-1}}$ by [Conjugacy in $\operatorname{SU}(2)$](/theorems/2475) (Step 1, part 1). Since $\chi_V$ is a class function, \begin{align*} \chi_V(t_z) = \chi_V(s t_z s^{-1}) = \chi_V(t_{z^{-1}}) = \sum_k m_k z^{-k}. \end{align*} Comparing coefficients of $z^k$: $m_k = m_{-k}$ for every $k \in \mathbb{Z}$. Hence $\chi_V|_T \in \mathbb{N}[z, z^{-1}]_{\mathrm{ev}}$, the additive monoid of Laurent polynomials in $z$ with non-negative integer coefficients invariant under $z \leftrightarrow z^{-1}$.[/step]

[guided]The restriction-to-torus argument uses two facts. First, every conjugacy class of $G = \operatorname{SU}(2)$ meets $T$ (this is part (4) of [Conjugacy in $\operatorname{SU}(2)$](/theorems/2475)), so a class function on $G$ is determined by its restriction to $T$. Second, the Weyl group $N_G(T)/T \cong \mathbb{Z}/2$ acts on $T$ via $t_z \mapsto t_{z^{-1}}$, and the conjugation invariance of $\chi_V$ forces the restriction $\chi_V|_T$ to be Weyl-invariant — equivalently, the Laurent polynomial coefficients satisfy $m_k = m_{-k}$. The decomposition $V|_T \cong \bigoplus_k \mathbb{C}^{m_k}$ uses the spectral theory of compact abelian groups: every continuous finite-dimensional representation of a compact abelian group is a direct sum of one-dimensional representations (characters), and the characters of $S^1$ are exactly the maps $z \mapsto z^k$ for $k \in \mathbb{Z}$. The non-negative integer $m_k$ is the multiplicity with which $z^k$ appears in $V|_T$.[/guided]

[step:Show the characters $\{\chi_n\}_{n \geq 0}$ form a $\mathbb{Q}$-basis of $\mathbb{Q}[z, z^{-1}]_{\mathrm{ev}}$] By [Irreducibility of $V_n$](/theorems/2476), the representations $V_n$ are irreducible for every $n \geq 0$. The character of $V_n$, restricted to $T$, is \begin{align*} \chi_n(z) := \chi_{V_n}(t_z) = z^n + z^{n-2} + \cdots + z^{-n} = \sum_{j=0}^n z^{n - 2j}. \end{align*} (This is the trace of $\rho_n(t_z) = \mathrm{diag}(z^n, z^{n-2}, \ldots, z^{-n})$ acting on $V_n$ in the monomial basis.) Each $\chi_n \in \mathbb{Z}[z, z^{-1}]_{\mathrm{ev}} \subseteq \mathbb{Q}[z, z^{-1}]_{\mathrm{ev}}$. [claim:The set $\{\chi_n : n \geq 0\}$ is a $\mathbb{Q}$-basis of $\mathbb{Q}[z, z^{-1}]_{\mathrm{ev}}$] [proof] Let $V := \mathbb{Q}[z, z^{-1}]_{\mathrm{ev}}$. The set $\{z^n + z^{-n} : n \geq 1\} \cup \{1\}$ is a $\mathbb{Q}$-basis of $V$, since every $f \in V$ with $f = \sum_k c_k z^k$ and $c_k = c_{-k}$ can be written uniquely as $c_0 + \sum_{n \geq 1} c_n (z^n + z^{-n})$. So $\dim_\mathbb{Q} V$ has a countable basis indexed by $\{0, 1, 2, \ldots\}$. Define a $\mathbb{Q}$-linear map $\Phi: \mathbb{Q}^{(\mathbb{Z}_{\geq 0})} \to V$ sending the basis vector $e_n$ to $\chi_n$. We show $\Phi$ is invertible by displaying its matrix as upper-triangular with $1$s on the diagonal in the basis $\{1, z + z^{-1}, z^2 + z^{-2}, \ldots\}$ on the codomain side. For $n = 0$: $\chi_0 = 1$. For $n \geq 1$: \begin{align*} \chi_n = z^n + z^{n-2} + \cdots + z^{-n} = (z^n + z^{-n}) + (z^{n-2} + z^{-(n-2)}) + \cdots \end{align*} The terms continue down to either $z + z^{-1}$ (if $n$ is odd) or $1$ (if $n$ is even, with the central term $z^0 = 1$ counted once). Explicitly: \begin{align*} \chi_n = \begin{cases} \sum_{j=0}^{n/2 - 1} (z^{n - 2j} + z^{-(n - 2j)}) + 1 & n \text{ even}, \\ \sum_{j=0}^{(n-1)/2} (z^{n - 2j} + z^{-(n - 2j)}) & n \text{ odd}. \end{cases} \end{align*} The leading term $z^n + z^{-n}$ appears with coefficient $1$. Lower terms $z^k + z^{-k}$ for $k < n$ of the same parity as $n$ appear with coefficient $1$; terms of opposite parity do not appear. Form the infinite matrix $A$ with $A_{n, k}$ the coefficient of $z^k + z^{-k}$ (or $1$ if $k = 0$) in $\chi_n$. Then $A_{n, n} = 1$ for all $n$, and $A_{n, k} = 0$ for $k > n$. So $A$ is upper-triangular with unit diagonal, hence invertible. Equivalently: the change-of-basis matrix from $\{\chi_n\}$ to $\{z^n + z^{-n}\} \cup \{1\}$ is upper-triangular with $1$s on the diagonal, so $\{\chi_n\}$ is a $\mathbb{Q}$-basis. [/proof] [/claim] [/step]

[step:Decompose $\chi_V$ in the basis $\{\chi_n\}$ and clear denominators] By Steps 1–2, $\chi_V|_T \in \mathbb{N}[z, z^{-1}]_{\mathrm{ev}} \subseteq \mathbb{Q}[z, z^{-1}]_{\mathrm{ev}}$. By the basis property, \begin{align*} \chi_V|_T = \sum_{n = 0}^{N} a_n \chi_n \end{align*} for some $N \geq 0$ and unique $a_n \in \mathbb{Q}$ (only finitely many nonzero by finite-dimensionality of $V$). Multiply by the common denominator: there exists $D \in \mathbb{Z}_{\geq 1}$ such that $D a_n \in \mathbb{Z}$ for every $n$. Write $D a_n = b_n^+ - b_n^-$ with $b_n^\pm \in \mathbb{Z}_{\geq 0}$ (positive and negative parts). Then \begin{align*} D \chi_V|_T + \sum_n b_n^- \chi_n = \sum_n b_n^+ \chi_n. \end{align*} The left-hand side is the character of $V^{\oplus D} \oplus \bigoplus_n V_n^{\oplus b_n^-}$ — a direct sum of $G$-representations with non-negative multiplicities $D, b_n^-$. The right-hand side is the character of $\bigoplus_n V_n^{\oplus b_n^+}$, also a $G$-representation. Both sides are characters of finite-dimensional continuous representations of $G$ with the same value on $T$, hence the same as class functions on $G$ (since class functions are determined by torus restrictions, by [Conjugacy in $\operatorname{SU}(2)$](/theorems/2475)). [/step]

[step:Apply unique decomposition to force $V \cong V_n$ for some $n$]By [Maschke's Theorem for Compact Groups](/theorems/2473), every finite-dimensional continuous representation of $G$ is completely reducible. Equivalently, the isomorphism class of a representation is determined by the multiset of irreducible summands. Equivalently, the characters of distinct irreducibles are linearly independent over $\mathbb{Q}$ (and indeed over $\mathbb{C}$, by [Schur's Lemma](/theorems/2414) and the orthogonality relations). [claim:Characters of pairwise non-isomorphic irreducible representations of $G$ are linearly independent] [proof] Suppose $W_1, \ldots, W_r$ are pairwise non-isomorphic finite-dimensional continuous irreducible representations of $G$ and $\sum_{j=1}^r \alpha_j \chi_{W_j} = 0$ for some $\alpha_j \in \mathbb{C}$. Form the orthogonality pairing on continuous class functions on $G$ (using the [Haar measure](/theorems/2472), which exists since $G$ is compact): \begin{align*} \langle f, h \rangle := \int_G f(g) \overline{h(g)} \, d\mu_G(g). \end{align*} By the orthogonality relations for characters of compact groups (a consequence of [Schur's Lemma](/theorems/2414) applied to the projection $V \otimes W^* \to V \otimes W^*$ averaged over $G$), $\langle \chi_{W_i}, \chi_{W_j} \rangle = \delta_{ij}$. Pairing $\sum_j \alpha_j \chi_{W_j} = 0$ with $\chi_{W_i}$ gives $\alpha_i = 0$ for every $i$. [/proof] [/claim] Now we exploit linear independence. From Step 3, \begin{align*} \chi_{V^{\oplus D} \oplus \bigoplus_n V_n^{\oplus b_n^-}} = \chi_{\bigoplus_n V_n^{\oplus b_n^+}}. \end{align*} Since both sides are direct sums of irreducibles (the $V_n$ by [Irreducibility of $V_n$](/theorems/2476), and $V$ by hypothesis) and characters of non-isomorphic irreducibles are linearly independent, the multisets of irreducible summands on each side coincide. In particular, $V$ appears on the left with multiplicity $D \geq 1$. So $V$ must appear on the right. Every summand on the right is some $V_n$. Hence $V \cong V_n$ for some $n \geq 0$.[/step]

[guided]The crux is the linear independence of the irreducible characters, which is a direct consequence of the orthogonality relations on a compact group $G$. The orthogonality relations themselves follow from [Schur's Lemma](/theorems/2414): for irreducibles $W, W'$, the matrix coefficient pairings $\int_G \pi_W(g)_{ij} \overline{\pi_{W'}(g)_{k\ell}} \, d\mu_G(g)$ vanish unless $W \cong W'$, and within $W \cong W'$ they reduce to a normalisation constant times $\delta_{ik} \delta_{j\ell}$. Tracing over $i = j$ and $k = \ell$ gives $\langle \chi_W, \chi_{W'} \rangle = \delta_{[W] = [W']}$. The denominator-clearing step turns a $\mathbb{Q}$-linear identity $\chi_V = \sum a_n \chi_n$ into a $\mathbb{Z}$-linear identity that is, after rearrangement, an equality of characters of *actual* representations. We need this because Maschke / linear independence of characters is a statement about characters of representations, not about formal $\mathbb{Q}$-combinations of them. The conclusion $V \cong V_n$ uses uniqueness of irreducible decomposition: since characters of non-isomorphic irreducibles are linearly independent, the decomposition of any representation into a direct sum of irreducibles is determined by its character. Comparing the two sides gives that $V$ — appearing on the left with multiplicity $D$ — must be one of the $V_n$ on the right. This completes the classification: $\{V_n : n \geq 0\}$ is a complete list of irreducible representations of $\operatorname{SU}(2)$ up to isomorphism.[/guided]