# fa.functional analysis – Are there references to functional variants of the Unbounded Knapsack Problem?

Looking for a version of the following problem, extended to solutions in $$ell^{infty}(mathbb{N})$$

Unbounded Knapsack Problem

$$max_{x_1,…,x_n} sum_{i=1}^n v_ix_i$$

$$text{ subject to }$$

$$sum_{i=1}^n w_ix_i leq W$$

$$w_i,v_i in mathbb{R}^+ ;;forall i$$

$$x_i in mathbb{N}$$

What I’d like to examine is the extended version

“Functional” Unbounded Knapsack Problem

$$max_{x_1,…,x_n} langle v,x rangle$$

$$text{ subject to }$$

$$langle w,x rangle leq W$$

$$w,v in ell^{infty}:w_i,v_i > 0;forall i in mathbb{N}$$

$$x in mathbb{N}^{infty}$$

I’m having a hard time finding any existing literature on this problem (or its continuous cousin defined on $$L^{infty}$$).

More generally, is there something akin to Calculus of Variations (except we’re optimizing functionals defined on $$ell^{infty}$$). Is there a variant of the Euler-Lagrange equation for functionals in sequence spaces?