Is there only one way to define a norm from an inner product?

Question

Given an inner product $\langle,\rangle$, we can define a norm by $||x|| = \langle x,x \rangle^{\frac{1}{2}}$. My question is, are there other ways to derive a norm from an inner product space and if not, how can one prove it?

Answer 1

There are other ways, for example $||x|| = 2\sqrt{\langle x, x \rangle}$.

If you want to exclude all norms of the form $K \sqrt{\langle x, x \rangle}$, then you need to specify what you mean by "deriving from".