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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 addition5.3) and scalar multiplication5.5) 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.

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Linearity of the Inner Product
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Norm Properties