Norms and Inner Products Is it possible to form an inner product from a norm or only the other way around?  When do inner products not have norms?
 A: If a norm $||\cdot||$ comes from an inner product $||x|| := \langle x, x \rangle$, then it will satisfy the so-called "parallelogram law": we have
$$||x+y||^2 = \langle x + y, x + y \rangle = \langle x, x \rangle + \langle x, y \rangle + \langle y, x \rangle + \langle y, y \rangle$$
and 
$$||x-y||^2 =  \langle x - y, x - y \rangle = \langle x, x \rangle - \langle x, y \rangle - \langle y, x \rangle + \langle y, y \rangle$$
so, adding these together, we get
$$||x+y||^2 + ||x-y||^2 = 2||x||^2 + 2||y||^2.$$
There are norms that do not satisfy this identity, though, so they cannot come from inner products.  For instance, the $L^p$ spaces where $p \neq 2$ have norms which do not satisfy the parallelogram law.
I believe (don't quote me on it) that satisfying the parallelogram law is equivalent to coming from an inner product, but the proof may be tricky.
A: Given an inner product you may always define a norm. Say $\langle \cdot,\cdot\rangle$ is an inner product on $V$ then $\Vert x\Vert^{2}=\langle x,x\rangle$. This turns out to always be real and it will define a norm. If you are given a norm, then you can sometimes define an inner product like the user above said. 
