Mathematically, what we are working with so far is called a Banach space, which is a normed linear vector space. To summarize, we defined our vectors as any list of real or complex numbers which we interpret as coordinates in the -dimensional vector space. We also defined vector addition (§5.3) and scalar multiplication (§5.5) in the obvious way. To have a linear vector space (§5.7), it must be closed under vector addition and scalar multiplication (linear combinations). I.e., given any two vectors and from the vector space, and given any two scalars and from the field of scalars , the linear combination must also be in the space. Since we have used the field of complex numbers (or real numbers ) to define both our scalars and our vector components, we have the necessary closure properties so that any linear combination of vectors from lies in . Finally, the definition of a norm (any norm) elevates a vector space to a Banach space.
Linearity of the Inner Product