### 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 ,

*additive*in its second argument,

*i.e.*,

*conjugate homogeneous*(or

*antilinear*) in its second argument, since

*is*strictly linear in its second argument with respect to

*real*scalars and :

*bilinear operator*in that context.

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Norm Induced by the Inner Product

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Banach Spaces