Let $F:L^2([0,1])\rightarrow \mathbb{R}$ be a convex functional. Consider the minimization problem \begin{align} \underset{f(\cdot) \in L^2([0,1])}{\min} F(f)\,\,\text{ subject to } \|f(\cdot) \|_2\leq \lambda, \end{align} with $\lambda>0$ and $\|f(\cdot) \|_2=\int_0^1 f\left(t\right)^2\text{d} t $. Are there some kind of KKT conditions for the minimizer of $F$?
You can even consider $F$ as linear if that‘s easier.