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 ,
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.
Norm Induced by the Inner Product