= Unitary highest-weight termination
{title2=$\|(J_-)^\ell|j,j\rangle\|^2=\ell(2j-\ell+1)\|(J_-)^{\ell-1}|j,j\rangle\|^2$}
For Hermitian $J_3$ and adjoint <angular momentum ladder operators> $J_+^\dagger=J_-$ on a positive-definite inner-product space, a <highest-weight vector> of weight $j$ obeys $J_+|j,j\rangle=0$. The commutation relations give $J_+(J_-)^\ell|j,j\rangle=\ell(2j-\ell+1)(J_-)^{\ell-1}|j,j\rangle$. Taking <inner products> forces $2j$ to be a nonnegative integer: otherwise the first negative factor would give a negative squared norm. The norm first vanishes at $\ell=2j+1$, so the generated <unitary representation> has <dimension> $2j+1$. Without positivity an abstract highest-weight module can remain infinite-dimensional even at an integral <highest weight>.
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