Change of inner product on Hilbert space

Question

Let $(\mathcal{V},\langle\cdot,\cdot\rangle_1)$ be a Hilbert space. If we change the inner product, can we then say anything about if that is a Hilbert space as well, i.e. when is $(\mathcal{V},\langle\cdot,\cdot\rangle_2)$ also a Hilbert space? Is it enough that $\langle\cdot,\cdot\rangle_2$ defines an inner product on $\mathcal{V}$?

Answer 1

No, it is not automatic that the vector space endowed with another inner product will be complete.

For example, let $H$ be any infinite dimensional Hilbert space,let $B$ be a Hamel basis of $H$ and consider the inner product on $H$ for which $B$ is an orthonormal basis. Then with that inner product $H$ is not complete.