What's the relationship between Banach space and inner product space

Question

I know that Banach space is a special Hilbert space. Inner product space is a special normed space. Hilbert space is a complete inner product space. Banach space is a complete normed space. I'm wondering what is the relationship between Banach space and inner product space.

Answer 1

A Banach space is a complete metric space, so that it has a distance function $d(x,y)$ and every Cauchy sequence converges to an element of the space.

A Hilbert space is a complete vector space, with an inner product $<x,y>$ or $x \cdot y$, that defines angles: $$ cos(\theta) = \dfrac{\langle x,y\rangle}{||x|| ||y||} $$ which defines the norm as $$ ||x|| = \sqrt{\langle x,x\rangle}. $$ and distance as $$ ||x-y|| = \sqrt{\langle x-y,x-y \rangle}. $$ So in a Banach space, you can talk about distance and convergence but not direction, while a Hilbert space has all of those. You need a Hilbert space for generalizing things like orthogonality, characterizing solutions to minimum distance problems (by way of the parallelogram equality), projection for statistics like OLS, and often useful results in approximation.

For example, consider the difference between $\mathbb{R}^N$ with the norm $$ ||x|| = \sup_{i} |x_i| $$ and the Euclidean norm $$ ||x|| = \sqrt{\sum_{i=1}^N x_i^2} = \sqrt{x\cdot x}. $$ You can think of the first as a Banach space but not a Hilbert space, and the second as a Hilbert space. Imagine what you lose by analyzing $\mathbb{R}^N$ without angles.

Answer 2

Every inner product induces a norm, so an inner product space is a normed vector space. Banach spaces are complete, normed vector spaces. So, the relation between inner product spaces and Banach spaces is that they're both normed vector spaces.

However, (1) an inner product space might lack the completeness that a Banach space possesses and (2) a Banach space norm need not be induced by an inner product. The norm needs to satisfy the parallelogram law for this to be the case (see https://en.wikipedia.org/wiki/Parallelogram_law#Normed_vector_spaces_satisfying_the_parallelogram_law).