Although they are complete, they do not necessarily have an inner product.
Although they are complete, they do not necessarily have an inner product. The L P spaces are all examples of them. They are all vector spaces, moreover the norm in them can be used to define a metric. Usually they are infinite dimensional, as they are frequently used in functional analysis; however, finite ones do exist. Every Hilbert Space is, for ten points, one of these complete normed vector spaces named for a Polish mathematician.
Although they are complete, they do not necessarily have an inner product.
{Banach} Space