What is the relation between a Banach space and a Hilbert space? [closed]

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.

Hilbert spaces are a stricter subset of Banach spaces but they have the additional structure of an inner product which allows you to talk about orthonormal bases, unitary operators and so on. Example: Fourier transform theory is really beautiful on $L^2(\mathbb{R})$ but it is much more complicated on other Banach spaces because you don't have a notion of self-duality like in $L^2(\mathbb{R})$. You actually have to abstract a lot to define the Fourier transform on other $L^p$ spaces.

Hilbert spaces have an easier structure and are in a way (most often infinite dimensional) Euclidian spaces. However, many spaces of interest that are Banach spaces are not Hilbert spaces, hence they are important too.

To see if a Banach space is a Hilbert space it suffice to show that the norm satisfies the parallelogram law. In other words, if we have a Banach space $X$ such that $$\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2$$ for all $x,y\in X$ then $X$ is actually a Hilbert space.

To deduce how the norm looks like is a good exercise. In the real case (the complex case is similar) think of the expansions of $$\|x+y\|^2=\langle x+y,x+y\rangle$$ and $$\|x-y\|^2=\langle x-y,x-y\rangle.$$ The result is $$\langle x,y\rangle=\frac{\|x+y\|^2-\|x-y\|^2}{4}.$$

I don't think they are "better", per se. A Hilbert space is a very special type of Banach space - one which is meant to generalize familiar notions from $\mathbb{R}^n$. (For instance, you can quite naturally speak of when two vectors in Hilbert space are orthogonal).

In general, Hilbert spaces are "easier" to understand than general Banach spaces, and are usually a good place to start if you are learning the subject (For instance, try to see why the Hahn-Banach theorem is much simpler for Hilbert spaces)