How do I prove that a function is well defined? in my case, a continuous linear function.

Question

This is my problem: Let $X$ and $Y$ be two normed vectorial spaces and let $L: X \rightarrow Y$ be a linear continuous functional. Let's now define $L^*: Y^* \rightarrow X^*$ like the the function that to each $y^*\in Y^*$ it associates $L^*y^* = l_{y^*}$, where: $(\forall y^*\in Y^*)$$l_{y^*}: X\rightarrow \mathbb{R}$, $l_{y^*}(x) = y^*(Lx)$. So in order to prove that $L^*$ is well defined, what do I need to do? Would it be enough to prove that $l_{y^*}$ is a linear function in $X$ for all $y^*\in Y^*$ and then that $L^*$ is linear?