TITLE: What is the relation between a Banach space and a Hilbert space?
QUESTION [28 upvotes]: What is the relation between a Banach space and a Hilbert space?
I know that a Hilbert space has inner product (and so a norm), but a Banach space has 
a just norm.

REPLY [2 votes]: Since most of the time we deal with $\mathbb{R}^n$, you can say that one has advantage over the other in $\mathbb{R}^n$. So I consider $\mathbb{R}^n$ and explain why Hilbert spaces are more useful in practice:
Hilbert space is defined as a vector space $H$ equipped with an inner product $\langle \cdot,\cdot \rangle$, consequently this inner product induces a norm which we call it a metric, $d(\cdot,\cdot)$. Hence, it is the time to define a new vector space $M$ endowed with the mentioned metric $d(\cdot,\cdot)$ which is a metric vector space. Now if every Cauchy sequence in $M$ converges to a point in $M$ we have Banach space which is a complete metric space. In $\mathbb{R}^n$ the relation is as follows:

Keep in mind that in a vector space $M$, the only tool that we have is the metric $d(\cdot,\cdot)$, so we can talk about length of a vector or distance between vectors. 
However, when we have inner product that satisfies Parallelogram law, we also have an inner product through which we can define projection from one vector to another which is of interest in practice because we can talk about the angles. Using this concept, optimization problems such as minimum distance to a subspace can be defined as projection of that point onto a set.