14. A necessary condition for extremum of a functional.

Theorem 2: Let A  be a subset of a normed linear space V  and let

be a given functional. if an element y0 in A gives a local minimum for J  relative the norm ||.|| then

for all admissible variations h.

Remark: The solutions to the problem above is called extremals.

Proof: Define the function

This obviously a regular one-variable function

and a necessary condition for extreme values to be attained in the point  is that

that is, 

The proof is complete.