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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