Does the Chain Rule Apply to Derivatives of Inner Products?

[brydustin]

I am trying to take the derivative of an
inner product (in the most general sense
over L^2), and was curious if the
derivative follows the "chain rule" for
inner products.

i.e. Does D_y(<f,g>) = <D_y(f),g> + <f,D_y(g)>
where D_y is the partial derivative w.r.t. y.

So for example, IT IS TRUE that if f=x*y and g=sin(x*y)
and the inner product <f,g> = Integral(f#g, w.r.t. x,-Pi,+Pi), f# = complex conj. of f.
then the equality holds.
In other words, differentiating w.r.t. y and integrating w.r.t x the forumla holds.
It seems more trivial if the variable which is being differentiated & integrated is the same.
But is it true in general?
What if we are differentiating more abstract inner products (i.e. not necessarily integration).

 

[tiny-tim]

hi brydustin!

yes, the https://www.physicsforums.com/library.php?do=view_item&itemid=353" works for any binary combination, provided it is bilinear (linear in both variables) …

its proof only requires (f+df,g) to equal (f,g) + (df,g) ​

(and (f,g+dg) to equal (f,g) + (f,dg))