= Solution
For a finite-dimensional <Lie algebra representation> $V$, a <weight of a representation> relative to $\mathfrak h$ is a linear functional $\mu\in\mathfrak h^*$ for which the simultaneous eigenspace
$$
V_\mu=\{v\in V:H\cdot v=\mu(H)v\text{ for every }H\in\mathfrak h\}
$$
is nonzero. Its elements are <weight vectors>, and $\dim V_\mu$ is the <weight multiplicity>. The set of weights is $\boxed{\Lambda_V=\{\mu:V_\mu\ne0\}}$. For a complex <semisimple Lie algebra>, the Cartan action on a finite-dimensional module is simultaneously diagonalizable, giving $V=\bigoplus_{\mu\in\Lambda_V}V_\mu$. The set of weights and their multiplicities are distinct information: several basis vectors can occupy the same point of a <weight diagram>.
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