An inner product space and its proper closed subspace with trivial orthogonal complement

Question

I am looking to do the following:

Construct an inner product space $X$ (with inner product $\langle \cdot, \cdot \rangle$ and a proper, closed subspace $Y$ of $X$ such that $Y^\perp = \{0\}$, ie $\langle x, y \rangle = 0 \ \forall y \in Y \iff x = 0$.

I see that we need $X$ to be infinite dimensional, hence isomorphic to $\Bbb R^n$ with the standard dot product, and then clearly not possible, as a proper (closed) subspace has a smaller dimension.

If someone could give me a hint as to how to start this construction, then I'd be most grateful, as I'm fairly stuck beyond this! As always, please make sure that the hint is reasonably minor, ie doesn't give away too much - I still want to learn from this question, not just be told the answer!

The question is on a course in linear analysis.

Answer 1

Following Daniel Fischer's hints in comments, here is an example:

1. Begin with $\ell^2$, its one-dimensional subspace $V$ spanned by vector $v = (1,1/2,1/3,\dots)$, and its orthogonal complement $V^\perp$. Then $(V^\perp)^\perp = V$.
2. Intersect all of the above with the set $F$ of sequences that have only finitely many nonzero elements.
3. Outcome: $X = \ell^2\cap F$ is an inner product space; $Y = V^\perp\cap F$ is closed in $X$, and $Y^\perp = V\cap F = \{0\}$. Also, $Y$ is a proper subset of $X$ since it does not contain, say, $(1,0,0,\dots)$.