Subspaces $X$ and $Y$ of a Hilbert Space with $\dim X<\infty$ and $\dim X<\dim Y$. I am working through some problems in the book Analysis Now by Pedersen, and I came across this problem:
Suppose $H$ is a Hilbert space and $X$ and $Y$ are closed subspaces of $H$ with $\dim X<\infty$ and $\dim X<\dim Y$. Prove that $X^{\perp}\cap Y\not=\{0\}$. 
I noticed someone asked the same question here: Hilbert space, functional analysis
But, I'm unsure if the answer is correct since it is not clear to me why $Y=X\cap Y+X^{\perp}\cap Y$.
If $X\subset Y$, then I would know how to solve the problem, but I am having trouble with the general case. 

By changing $Y$ by a subspace of dimension $\dim(X)+1$, and $H$ by $X+Y$, we can assume $H$ is finite dimensional.
Now consider the canonical quotient map $T:Y\to H/X^\perp\simeq X$. By the rank-nullity theorem,
$$\operatorname{dim}(Y)=\operatorname{dim}(TY)+\operatorname{dim}(\ker T)\leq\operatorname{dim}(X)+\operatorname{dim}(Y\cap X^\perp)$$
But $\operatorname{dim}(Y)>\operatorname{dim}(X)$, so
$$0<\operatorname{dim}(Y\cap X^\perp).$$
