Suppose we have some functionals $H,G:l^p(\mathbb{N}^+)\to\mathbb{R}$, and we want to find some $p \in l^p(\mathbb{N^+})$ which minimize $H$, subject to the constraint that $G(p)=0$ is constant.
As an example, let $H(Y)$ be the entropy of a random variable $Y=0,1,2,3,\ldots$. What distribution on $Y$ will maximize $H(Y)$, while holding the mean fixed at $A$? $$ min~ \sum\limits_{n=0}^\infty p(Y=n)\log_2p(Y=n)$$ $$ s.t. ~ \sum\limits_{n=0}^\infty p(Y=n) = 1$$ $$ s.t. ~ \sum\limits_{n=0}^\infty np(Y=n) = A$$
I stumbled on this problem in an information theory text, where they treat it identically to the finite dimensional case and apply Lagrange multipliers. However, given that $l^p$ is infinite dimensional, I'm skeptical that this will always work. I suspect one could derive something like this using the calculus of variations and a weird measure, but I'm not sure. Is there a text I should read that considers this problem?
On a side note, does anyone have a good way of searching for $l^p$ spaces via google?
It is possible to deal with your problem as an infinite dimensional optimization problem.
Suppose $H : \ell^p \to \mathbb{R}$ is Fréchet-differentiable and $G : \ell^p \to \mathbb{R}^2$ is linear and bounded.
If $p \in \ell^p$ is a minimizer of $H$ over $G(p) = c$, then you will find a multiplier $\lambda \in \mathbb{R}^2$, such that \begin{equation*} H'(p) + G^\star \, \lambda = 0. \end{equation*} Here, $G^\star : \mathbb{R}^2 \to \ell^{p'}$ is the adjoint of $G$.
For your problem, however, you have to choose a different space than $\ell^p$, since the mean is not bounded w.r.t.\ the $\ell^p$-norm.