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Proving that two closed sets in Hilbert space are orthogonal

I would like to ask for your help in an exercise I’m having trouble with.
Let $H$ be a (Hilbert) space, $\{x_n\}_{n \in \mathbb N}$ a countable set of elements of $H$, $x \in H$, and $\{y_n\}_{n \in \mathbb N}$ a countable dense set of $H$.
Prove that if $y_n \perp x_n$ for all $n \in \mathbb N$ then $y \perp x$.
I have no idea how to tackle this problem, and the book only says that the answer is obvious.
If someone can provide me with a proof to this problem or another question that I can find an answer for, I would be very grateful.
Thanks in advance.

A:

Note that if $x\perp y$, you have $x+y^\perp=x+y=\{0\}$, so, since $H$ is a Hilbert space, $x+y$ is closed and bounded (because $x^\perp$ is dense), hence compact. If $x+y$ is not the whole space (this can’t happen, because $H$ is finite-dimensional), then $x+y=\{0\}$ and we have $x=y$.
So, if $x\perp y$, then $x\perp y$ in all of the finite-dimensional Hilbert spaces that arise as closed subspaces of $H$.

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[^1]: Politecnico di Torino, V. Corvaja 4, 10129 Torino, Italy, `{lorenzini, antonio.estob@polito.it}`.

[^2
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