Let $T$ be a compact torus, meaning a compact abelian Lie group isomorphic to $(S^1)^r$ for some integer $r\ge 0$, where $S^1:=\{z\in\mathbb C:|z|=1\}$. Let $V$ be a finite-dimensional complex [vector space](/page/Vector%20Space), let $GL(V)$ be the group of complex-linear automorphisms of $V$, and let
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\begin{align*}
\rho:T\to GL(V)
\end{align*}
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be a continuous [group homomorphism](/page/Group%20Homomorphism). A continuous character of $T$ means a continuous group homomorphism $\lambda:T\to S^1$. For every continuous character $\lambda:T\to S^1$, define the $\lambda$-weight space by
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\begin{align*}
V_\lambda:=\{v\in V:\rho(t)v=\lambda(t)v\text{ for every }t\in T\}.
\end{align*}
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Then the set
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\begin{align*}
\{\lambda:T\to S^1:\lambda\text{ is a continuous character and }V_\lambda\ne\{0\}\}
\end{align*}
where the [direct sum](/page/Direct%20Sum) is over continuous characters and equivalently over the finitely many continuous characters with nonzero weight space.