Let $H$ be a Hilbert space with norm $\|\cdot\|_1$ induced by the inner product. And if we define another norm on $H$, we will denote this norm $\|\cdot\|_2$.
Now we suppose that the two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent. Obviously, $(H,\|\cdot\|_2)$ is a Banach space.
My questions are: Can we have more? Can we say that $(H,\|\cdot\|_2)$ is also a Hilbert space?
(Can we always in this case define an inner product such that it's induced norm is $\|\cdot\|_2$ ?)