A Banach space is a vector space over the field of real numbers, or over the field of complex numbers, together with a norm. And in a Banach space the metric topology determine by the norm is complete. Thus the assertion that a certain object is a Banach space includes several different pieces of information.

We now describe a Banach space more carefully. We begin with a vector space X having either the real numbers or the complex numbers as its field of scalars, (see the article on abstract linear spaces.) By a norm on X we understand a function ∥ ∥: X [0, ) that satisfies the following three conditions:

• (1) ∥x∥ = 0 if and only if x = 0 is the zero vector in X,
• (2) ∥x∥ = || ∥x∥ for all scalars and all vectors x,
• (3) ∥x + y∥ ∥x∥ + ∥y&par...