A normed linear space can be (uniquely) completed to a Banach space. However, whereas a Banach space is a match for its practically unique norm, there are many possible norms that can be used in a linear space, and depending on a choice of norm we obtain many different completions of a single space. This phenomenon is discussed first in the case of the space of sequences that have all but a finite number of coordinates equal to zero, and in the case of the space of polynomials. The injective and projective tensor norms, which show up naturally in the tensor product of two simple sequence spaces, illustrate this principle further, but they have their own importance, reaching far beyond the scope of the book.