Prove $x \cap x ^ \perp =\left\{0\right\} $

Question

I'm learning about Hilbert Spaces and one of the definitions says that $ H=Y\oplus Y^{\perp}$. I'm not having problem with the proof of uniqueness. I'm trying to prove $Y \cap Y ^ \perp =\left\{0\right\} $ where $Y \subset H $ and $Y$ is closed.

My approach is the following:

$x \in Y \cap Y ^ \perp \implies x \in Y$ and $ x \in Y ^ \perp $

and since $Y ^ \perp=\left\{x\in H: x\perp Y\right\} $

Therefore $x\perp x \implies x = \left\{0\right\} $

Is my proof right or is there something I'm clearly missing?

Answer 1

$x\in Y\cap Y^{\perp}$ implies $<x,x>=0$ implies $x=0$