### Linearity of the Inner Product

Any function
of a vector
(which we may call an
*operator* on ) is said to be *linear* if for all
and
, and for all scalars and in
,

*additivity*:*homogeneity*:

*e.g.*, can be linear or not with respect to each of its arguments.

The inner product
is *linear in its first argument*, *i.e.*,
for all
, and for all
,

The inner product is also *additive* in its second argument, *i.e.*,

*conjugate homogeneous*(or

*antilinear*) in its second argument, since

The inner product *is* strictly linear in its second argument with
respect to *real* scalars and :

Since the inner product is linear in both of its arguments for real
scalars, it may be called a *bilinear operator* in that
context.

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Norm Induced by the Inner Product

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Banach Spaces