5
$\begingroup$

Let $V$ be a vector space and let $W$ be its subspace of infinite codimension. Let $\mathcal{F}_W$ be the family of all finite-rank operators on $V$ with range contained in $W$. Consider the left-sided ideal in $\mathcal{L}(V)$ generated by $\mathcal{F}_W$. Does this ideal contain all the finite-rank operators on $V$?

1 Answers 1