I am attempting to prove that the multivariate distribution with maximum entropy for a given covariance is a Gaussian. (PRML, Bishop, problem 2.14).

Bishop suggests the use of Lagrange multipliers - concretely, that should maximize $$ \text{H}[x] = -\int p(x)\log(x)dx $$ subject to the constraints \begin{align*} \int p(x)\,dx &= 1\\ \int xp(x)\,dx &= \mu\\ \int p(x)(x - \mu)(x - \mu)^T\,dx &= \Sigma. \end{align*}

Applying Lagrange multipliers to the maximization of a functional (as opposed to the maximization of a function on $\mathbb{R}^N$) is alien to me. If someone could direct me to a reference describing why Lagrange multipliers continue to "work" in infinite dimensional function spaces, I would appreciate it.