In the lecture my professor wrote that the standard inner product on $R^n$ is given by 

$\langle x, y \rangle = x^Ty = \sum\limits\_{i=1}^n x\_i y\_i$ 

which induces a norm $\sqrt{\langle x,x \rangle} = \|x\|\_2$

My question is do inner products induce other types of norms...or rather are norms such as the 1-norm or the $\infty$-norm induces by some inner product?