TITLE: Relationship between inner product and norm
QUESTION [36 upvotes]: 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?

REPLY [4 votes]: For the second question, to give a slightly more abstract point of view, if a space can be equipped with an inner product, as proven by @Ian, it can always be equipped with a norm. In other words, the set of all inner product spaces is a subset of the set of all normed vector spaces. Furthermore, the former is actually a strict subset of the latter since one cannot always induce an inner product given a norm. This "strict subset" relationship generalizes to more "spaces". Namely, the set of all normed vector spaces is a strict subset of the set of all metric spaces, where one can define a distance measure. And the set of all metric spaces is a strict subset of the set of all topological spaces, where one can define some of the most basic mathematical concepts such as convergence. 
You might have noticed that some spaces are "better" than others in the sense that they can be equipped with more concepts, which makes it easier to talk about some problems. For example, the concept of orthogonality comes from inner product being zero, but in a topological space, one cannot always define an inner product (since an inner product space is a strict subset of the set of all topological spaces, so when one has a topological space, that space may not be an inner product space), as a result, one may not be able to even talk about orthogonality in that space because it is not defined there. By the way, a Hilbert space (a complete inner product space) is basically THE space with all the nice properties and concepts.