Oh! Then it is easy to explain this to you: hilbert space is a vector space with inner product and it is complete with respect to the norm induced by the inner product. So, two features are completeness with respect to a norm, and the norm comes from an inner product. So, strip those off! So, a normed linear space is a vector space with a norm. If you ask the space to be complete with the norm, these are Banach spaces. Just words, I know.
But, if you think about them, you'll see a lot of nice things.