### 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 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*.

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Linearity of the Inner Product

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Norm Properties