# Hilbert space with two equivalent norms

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 $$\|\cdot\|_2$$ ?)

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.