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 (§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 $\underline{x}\in{\bf C}^N$ and $\underline{y}\in{\bf C}^N$ from the vector space, and given any two scalars $\alpha\in{\bf C}$ and $\beta\in{\bf C}$ from the field of scalars ${\bf C}^N$, the linear combination $\alpha \underline{x}+ \beta\underline{y}$ 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.