I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a gradient flow, we would be happy?!
I know the answer about the gradient of a function, but have no idea about the gradient of a functional!
Can anyone help me? Thanks for your attention!
A functional is just a function on a space of functions, so it is defined on an infinite dimensional space. The visualization is the same as in finite dimensions, but there are many more things that can go wrong. The functional must be suitably differentiable. You need an inner product which might be continuous in the function space topology, biut it might not generate the topology, so the gradient a priory lies in the Hilbert space completion of the functions space -- so it might point outside of the function space. Beyond Banach spaces ODE's behave differently, yuou might loose existence or uniqueness of solution curves. But often it works.
