Banach Spaces

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 $N$ real or complex numbers which we interpret as coordinates in the $N$-dimensional vector space. We also defined vector addition in the obvious way. It turns out we have to also define scalar multiplication, that is, multiplication of a vector by a scalar which we also take to be an element of the field of real or complex numbers. This is also done in the obvious way which is to multiply each coordinate of the vector by the scalar. To have a linear vector space, it must be closed under vector addition and scalar multiplication. That means given any two vectors $\underline{x}\in{\bf C}^N$ and $\underline{y}\in{\bf C}^N$ from the vector space, and given any two scalars $c_1\in{\bf C}$ and $c_2\in{\bf C}$ from the field of scalars, then the linear combination $c_1 \underline{x}+ c_2\underline{y}$ must also be in the space. Since we have used the field of complex numbers ${\bf C}$ (or real numbers ${\bf R}$) to define both our scalars and our vector components, we have the necessary closure properties so that any linear combination of vectors from ${\bf C}^N$ lies in ${\bf C}^N$. Finally, the definition of a norm (any norm) elevates a vector space to a Banach space.