Continuous limit of discrete position basis

Question

Say we have a $1D$ lattice with spacing $a$ between two sites. How does one formally map the discrete position basis of the lattice to a continuous one in the limit $a\to 0$.
For instance how does quantities such as $\lvert i\rangle$ with $i=xa$ maps to $\lvert x \rangle $? Or the orthogonality relations $\langle i \lvert j\rangle=\delta_{ij}$ and $\langle x \lvert x'\rangle=\delta(x-x')$?

Answer 1

It was shown by the German mathematician Cantor that any mapping

$$ f:\mathbb N \rightarrow \mathbb R $$

cannot be surjective, which means that there is no way to map a discrete infinite basis in a Hilbert space into a continuous one. This is rephrased as: a separable Hilbert space cannot be isomorphic to a nonseparable one. This means that there is a negative answer to all your questions.

Answer 2

I think we can try to formalize it: Embed $L^2(\mathbb N)$ into $L^2(\mathbb R)$ by having a constant value on each interval $[n \epsilon, (n+1)\epsilon)$. Then let $\epsilon \rightarrow 0$.

We get a sequence of subspaces. A vector should be approximated increasingly well by its projection to each subspace as $\epsilon \rightarrow 0$.