If you have spent some time with undergraduate mathematics, you would have probably heard the word “norm”. This term is encountered in various branches of mathematics, like (as per Wikipedia):

But, it seems to occur only in abstract algebra. Although the definition of this term is always algebraic, it has a topological interpretation when we are working with vector spaces. It secretly connects a vector space to a topological space where we can study differentiation (metric space), by satisfying the conditions of a metric. This point of view along with an inner product structure, is explored when we study functional analysis.

Some facts to remember:

- Every vector space has a norm. [Proof]
- Every vector space has an inner product (assuming “Axiom of Choice”). [Proof]
- An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space. [Proof]
- All norms are equivalent in finite dimensional vector spaces. [Proof]
- Every normed vector space is a metric space (and NOT vice versa). [Proof]
- In general, a vector space is NOT same a metric space. [Proof]