Does boundedness in topological vector space imply boundedness in subspace

Question

Let $X$ be a topological vector space, $Y$ be a subspace. If $M\subset Y$ is bounded in $X$ (i.e. for every open set $U$ of $X$, there exists $\alpha$,s.t. if $\left|\beta\right|>\left|\alpha\right|$ $M\subset \beta U$), is it true that $M$ is bounded in $Y$?