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

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).
