I've known the norm (aka. unit vector in the direction of) a vector (let's call it $v$) to be given by the formula: $v/\sqrt{v \cdot v}$, where $\cdot$ is the dot product.
But can this be generalized to inner products, such that the norm the norm of a vector can be found by $v/\sqrt{< v,v>}$, where $<v,v>$ is the vector's inner product with itself? (since the dot product is an inner product this definition would agree with the previous one)
Any help is appreciated.