  In this paper convex optimization techniques are employed for
  convex optimization problems in infinite dimensional Hilbert spaces. A first
order optimality condition is given. Let $f : \mathbb{R}^{n}\rightarrow
\mathbb{R}$ and let $x\in \mathbb{R}^{n}$ be a local solution to the problem
$\min_{x\in \mathbb{R}^{n}} f(x).$ Then $f'(x,d)\geq 0$ for every direction
$d\in \mathbb{R}^{n}$ for which $f'(x,d)$ exists. Moreover, Let $f :
\mathbb{R}^{n}\rightarrow \mathbb{R}$ be differentiable at $x^{*}\in
\mathbb{R}^{n}.$ If $x^{*}$ is a local minimum of $f$, then $\nabla f(x^{*}) =
0.$ A simple application involving the Dirichlet problem is also given.
