If $X$ is a Banach space any capable conditions to make sure that $X$ is also Hilbert?

Question

Let $X$ be a Banach space and suppose that $X$ is isometric with a Hilbert space.Is it true to say that $X$ is also a Hilbert space?If I assume that $X$ is isomorphic with a Hilbert space is it still true?

Answer 1

YES for the first question, and NO for the second. If $T$ is an isometry from $X$ and $Y$ then $ \langle Tx, Ty \rangle=\langle x, y \rangle$ gives an inner product on $Y$ which makes it a Hilbert space. $\mathbb R^{2}$ with any norm is isomorphic to $\mathbb R^{2}$ with the Euclidean norm . Take the norm $\|(x,y)\|=|x|+|y|$ for a counter-example to the second statement. I leave it to you to verify that this norm does not satisfy Parallelogram Law which means that the space is not an inner product space. For Parallelogram Law see https://en.wikipedia.org/wiki/Parallelogram_law