Let $\mathfrak g$ be a Lie algebra and $U(\mathfrak g)$ is the universal enveloping algebra of $\mathfrak g.$ Let $V$ be a vector space is a unital $U(\mathfrak g)$ module. I want to show that the map $V\to V$, $v\mapsto Xv$ is linear where $X\in\mathfrak g$. I can easily show that $X(v+w)=Xv+Xw$ which follows from definition. But now how to show that $X(cv)=cXv$ for any scalar $c\in\mathbb C.$

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    $\begingroup$ What exactly is your definition of what it means for a vector space to be a $U(\mathfrak{g})$-module? By some definitions, $X(cv)=cXv$ would simply be part of the definition. In any case, the definition must include some compatibility between the scalar multiplication of $V$ and the modulle structure, and you should be able to use that compatibility to deduce that $X(cv)=cXv$. $\endgroup$ – Eric Wofsey May 19 at 16:05

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