Non continuously differentiable but inner product finite

[Sumanta]

Hello,

I was trying to understand Green's function and I stumbled across the following statements which is confusing to me.

I was referring to the following site

http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node79.html

Here the author says the following

"What if $ u$ is not a continuously differentiable function? Then its image $ Lu$ is not square-integrable, but the inner product <v, Lu> is still well-defined because it is finite. For example, if u is a function which has a kink, then $ Lu$ would not be defined at that point and $ Lu$ would not be square-integrable. Nevertheless, the integral of $ \overline v Lu$ would be perfectly finite."

I don't understand the fact is if Lu is not defined how can u define an inner product with v at any point, ie <v, Lu>. What does it mean physically at all, is it a mathematical jugglery to move the L operator to v and then say that look it is still defined? I am totally confused.

Thanks a lot for any help in advance.

 

[HallsofIvy]

Why do you say "Lu is not defined"? If Lu is not square-integrable, then it is not in L² but it is in some larger space, of which L² is a subspace. The innerproduct can be defined in that larger space.

 

[Sumanta]

Hi,

I say L is not defined because of the following. Let's give an example. Since L can be d^2/dx^2 + a(x) d/dx + b(x) and if u consider the function u s.t

for say (a< x <b), a<0, b >0

u(x) = 0 x<0,
= x x>= 0

The fn u is cont but is not differentiable at x = 0. So I am not sure how for such functions u can define the operator like this. This is my question.

Regards
Sumanta