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 $V$?
Extensions of finite-rank operators
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linear-algebra
vector-spaces
operator-theory