functional analysis

1. Suppose the real-valued functional defined on an open subset D of a normed space has a

relative minimum at . Show that if is twice Gateaux differentiable at , then

⟨ ( ) ⟩ for all .

2. Let be a real-vlued functional defined on an open region D in a normed space . Suppose that

at the first Frechet differential vanishes identically on within a sphere, ( ) ( )

exists and is continuous in ; and the lower bound of ⟨ ( ) ⟩ for ‖ ‖ is positive. Show

that obtains a relative minimum at