Let $X$ be a topological vector space and $Y$ a linear subspace of $X$. The closure of $Y$, say $\overline{Y}$, is also a linear subspace of $X$. To show that, let $x, y \in \overline{Y}$ and $a, b \in \mathbb{C}$. If $ax + by \in \overline{Y}$, then $\overline{Y}$ is a linear subspace. Since $\overline{Y}$ is closure of $Y$, there exist $x_n, y_n \in Y$ such that $x_n \to x$ and $y_n \to y$ as $n \to \infty$. Then $ax_n + by_n \in Y$ due to the linear subspace $Y$. And $ax_n + by_n \to ax + by$ due to the continuity of topological vector space $X$. Thus $ax + by \in \overline{Y}$, proved.
Can you check if my proof is correct? Thank you!