Can a pre-Hilbert space be separable?

Question

I am confused about separability and completeness and I would like to know if there exists a non complete inner product space that is separable?

Thanks

Answer 1

Consider the space $C_c(\mathbb N)$, the space consisting of (say complex valued) functions with finite support. It is a pre-Hilbert space with inner product given by $$ \langle f,g\rangle = \sum_{\mathbb N} f(n)\overline{g(n)} $$ This space is separable but not complete.

Answer 2

As every subset of a separable space is separable, you can take any dense proper subspace of a separable hilbert space. For example $C^1([0,1])$ is a dense subspace of $L^2([0,1])$. As $L^2([0,1])$ is separable it follows that $C^1([0,1])$ is separable too, but it is not complete with respect to the $L^2$-norm.