Suppose we have some vector space $V$ for which we have defined an inner product $\langle \cdot\rangle$. Thus we have an inner product space. Is it true that $\forall x \in V : \lVert x\rVert := \sqrt{\langle x,x\rangle}$ is the only possible norm in $V$?
Let's consider the following example, that shows, what happens if it is not true (if by an inner product induced norm is not the only possible in this vector space):
Let $\mathcal{P}$ be a vector space of polynomials with an inner product defined as $$\langle P,Q\rangle=\int_0^1P(x)Q(x)\operatorname{d}x \quad \forall P,Q\in\mathcal{P}$$ Suppose we want to find an angle $\phi$ between two polynomials $P_1$ and $P_2$. We use a standard formulae for an angle between two vectors $$\cos\phi={\langle P_1,P_2\rangle \over \rVert P_1\lVert\rVert P_2\lVert}$$ So the norms in denominator should be the ones induced by the inner product, otherwise the angle will depend on how we define a norm.
So one more time: I suspect, that for a vector space with defined inner product, the norm of a vector, that lies in this vector space is also defined as induced by the inner product and is unambiguous. Is it true?