Let $G$ be a compact connected Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $T\le G$ be a maximal torus with Lie algebra $\mathfrak t$, and let $X^*(T):=\operatorname{Hom}_{\mathrm{cts}}(T,U(1))$ be the character group of $T$. Let $\mathfrak g_{\mathbb C}:=\mathfrak g\otimes_{\mathbb R}\mathbb C$ and $\mathfrak t_{\mathbb C}:=\mathfrak t\otimes_{\mathbb R}\mathbb C$. Let $R\subset X^*(T)$ be the root system of $\mathfrak g_{\mathbb C}$ with respect to $\mathfrak t_{\mathbb C}$, identifying each root character with its complex-linear differential on $\mathfrak t_{\mathbb C}$. Choose a positive root system $R^+\subset R$, and let $\Delta\subset R^+$ be the corresponding set of simple roots. For each $\alpha\in R$, let $\alpha^\vee$ denote its coroot. Define $\Lambda^+(G,T,R^+)$ to be the set of all $\lambda\in X^*(T)$ such that $\langle \lambda,\alpha^\vee\rangle\in\mathbb Z_{\ge 0}$ for every $\alpha\in\Delta$. For every continuous irreducible finite-dimensional complex representation $\rho:G\to GL(V)$, the restricted representation $\rho|_T$ has a finite $T$-weight set in $X^*(T)$, ordered by declaring $\mu\le \nu$ when $\nu-\mu$ is a finite $\mathbb Z_{\ge 0}$-linear combination of elements of $R^+$. With respect to this order, $\rho$ has a unique highest weight $\lambda_\rho\in\Lambda^+(G,T,R^+)$. The assignment $[(\rho,V)]\mapsto \lambda_\rho$ defines a bijection from the set of isomorphism classes of continuous irreducible finite-dimensional complex representations of $G$ onto $\Lambda^+(G,T,R^+)$.