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:

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.,


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