Is it possible to extend the inner product from an incomplete inner product space E onto H ,such that H would become a Hilbert space?

Question

Let E be an incomplete inner product space. Let H be the completion of E . Is it possible to extend the inner product from E onto H such that H would become a Hilbert space? I think the answer is to establish with continuity of the inner product but do not know how to show?

Answer 1

We can use the fact that every uniformly-continuous function defined on a dense subset extends continuously onto a complete metric space. A non-complete metric space is a dense subset of its completion, and the inner-product is uniformly-continuous on the "pre-complete" space.