# Hilbert space, functional analysis

Let $X$ and $Y$ be closed subspaces of a Hilbert space $H$. Assume that dim $X < \infty$, and dim $X$ < dim $Y$. Show that $X^{\perp} \cap Y \neq \{0\}$.

I want to proof it by contradiction. But I don't know how to use dim $X$ < dim $Y$.