I understand that there can be many different types of norms (e.g. mean norm, Cartesian norm, supremum norm etc). Are there also other types of inner products apart from $\langle x,y \rangle= \sum_{j =1}^n x_j y_j$? Also, I read that for any inner product on a vector space V the function $\|x\| = \sqrt{\langle x,x \rangle}$ defines a norm on the vector space. Why is that so? So does this formula work for all the different inner products?
Inner Products – Relationship Between Inner Product and Norm
inner-products
Best Answer
Yes, there are many different types of inner products. Consider the inner product on $L^2$ given by $\langle f, g \rangle = \int f(x) \overline{g(x)} dx$.
An inner product $\langle , \rangle$ always defines a norm by the formula $||x||^2 = \langle x, x \rangle$. You can check that all the conditions of a norm are satisfied. However, the converse is not true, that is, not every norm gives rise to an inner product. Norms which satisfy the parallelogram law can be used to define inner products via the polarization identity.