In order for a product to be defined, there must be some additional structure in the space. It has to be either a Hilbert space (with an inner product), or a Banach algebra (with a multiplicative structure).
Also, as HallsofIvy has pointed out, a Banach space is not the same as a normed linear space.
One more time and then we give up!
A "Banach space" is a vector space having a norm such that "Cauchy sequences" converge.
While "scalar multiplication" and "vector addition" are defined in any vector space, multiplication of vectors is NOT generally defined. There is no "standard multiply"! It is impossible to answer your question with knowing how you are defining "XnYn".