Hilbert space with two equivalent norms

Question

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 $\|\cdot\|_2$ ?)

Answer 1

A norm equivalent to an inner-product norm need not itself be an inner-product norm.

This can be seen in finite dimensions, say $\mathbb{R}^2$, where the 2-norm and 1-norm are equivalent, but only the former is induced by an inner product.