Does the inner product $\langle \cdot, \cdot \rangle$ induce any other norms other than the 2 norm? 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?

Every inner product on $\mathbb{R}^n$ can be written as $\langle x, y \rangle = x^t A y$, where $A$ is a (symmetric) positive definite matrix. These matrices can be orthogonally diagonalized, i.e. there is an orthogonal matrix $M$ so that $A = M^t \operatorname{diag}(\lambda_1, \ldots, \lambda_n) M$. This means in particular $\langle x, y \rangle = (Mx)^t \operatorname{diag}(\lambda_1, \ldots, \lambda_n) (Mx)$.
Now note that the open balls with respect to the inner product $\langle x, y \rangle = x^t \operatorname{diag}(\lambda_1, \ldots, \lambda_n) y$ are ellipsoids and the map $x \mapsto Mx$ is essentially the composition of reflections and rotations [of course this can only be visualized for $n \le 3$]. So the open balls of an arbitrary inner product in $\mathbb{R}^n$ are rotated and reflected ellipsoids.
