= Solution
We consider finite-dimensional continuous complex representations of <SU(2)>. Its maximal <torus> is $\operatorname{diag}(e^{i\theta},e^{-i\theta})$. The <irreducible representations> are
$$
\boxed{V_n=\operatorname{Sym}^n(\mathbb C^2),\qquad n=0,1,2,\ldots,\qquad \dim V_n=n+1.}
$$
They may be realized as homogeneous <polynomials> of degree $n$ in two variables, with the induced linear substitution action. Their <torus> weights are $n,n-2,\ldots,-n$, each once. The centre $-I$ acts by $(-1)^n$, so precisely the even-$n$ representations descend to <SO(3)>. In physics $n=2j$ labels integral or half-integral spin $j$.
Here is a classification argument. Average any positive definite Hermitian form over the compact group using normalized <Haar measure>. The averaged form is invariant, so the orthogonal complement of an <invariant subspace> is invariant. Repeatedly splitting gives complete reducibility. Differentiate an <irreducible representation> and complexify to the <Lie algebra> $\mathfrak{sl}_2$, with generators $H,E,F$ satisfying $[H,E]=2E$, $[H,F]=-2F$, and $[E,F]=H$. <Torus> periodicity gives integer $H$-weights. Choose a vector $v$ of maximal weight $n$; then $Ev=0$. Induction on $k$ using the commutators gives
$$
EF^kv=k(n-k+1)F^{k-1}v.
$$
The chain terminates: if $F^{p}v\ne0$ and $F^{p+1}v=0$, the displayed formula forces $n=p\geq0$. The vectors $v,Fv,\ldots,F^nv$ have distinct weights, span an invariant subrepresentation, and hence span the <irreducible representation>. Their generator action agrees with $\operatorname{Sym}^n\mathbb C^2$. Invariance under the <Lie algebra> is equivalent to invariance under the connected group, completing the classification. Conversely, a nonzero <invariant subspace> of a <symmetric power> contains a weight vector by projection onto <torus> weights. Raising it to the <highest weight> and then lowering generates every monomial, proving that the <symmetric power> is irreducible.
The character on the <torus> is
$$
\chi_n(\theta)=\sum_{k=0}^n e^{i(n-2k)\theta}
=\frac{\sin((n+1)\theta)}{\sin\theta},
$$
with the endpoints interpreted by continuity. Every representation has a unique decomposition $V\cong\bigoplus_{n\geq0}m_nV_n$. Its multiplicities can be computed from character inner products,
$$
\boxed{m_n=\int_{\mathrm{SU}(2)}\chi_V(g)\overline{\chi_n(g)}\,dg
=\frac2\pi\int_0^\pi\chi_V(\theta)\chi_n(\theta)\sin^2\theta\,d\theta.}
$$
The sine orthogonality directly verifies orthonormality of the irreducible characters. Alternatively, if $w_k$ is the dimension of the weight-$k$ subspace, the weight lists give $w_n=m_n+m_{n+2}+\cdots$, so \b[$m_n=w_n-w_{n+2}$] for $n\geq0$. This gives a constructive decomposition from <torus> weights. In particular, multiplying weight characters yields the <Clebsch-Gordan decomposition> $V_m\otimes V_n\cong V_{m+n}\oplus V_{m+n-2}\oplus\cdots\oplus V_{|m-n|}$. Arbitrary unitary Hilbert-space representations of this compact group similarly decompose into Hilbert <direct sums> of these finite-dimensional irreducibles; the finite-dimensional statement above is the standard representation-theory setting.
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