A subset of a linear space $X$ is called infinite-dimensional if it is not contained in a finite-dimensional linear subspace of $X$.
Problem. Let $L$ be an infinite-dimensional subset of the linear space $\mathbb R^\omega$. Is there an infinite set $I\subseteq\omega$ such that for every infinite set $J\subseteq I$ the projection of $L$ to $\mathbb R^J$ is infinite-dimensional?
Remark. The answer to this problem is affirmative if every function $f\in L$ has finite range.