# Article

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Keywords:
$u$-ideals; finite rank; compact; and weakly compact operators; Hahn-Banach extension operators
Summary:
Let $X$ be a Banach space. We give characterizations of when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal W}(Y,X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when ${\cal F}(X,Y)$ is a $u$-ideal in ${\cal W}(X,Y)$ for every Banach space $Y$, when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal W}(Y,X^{**})$ for every Banach space $Y$, and when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal K}(Y,X^{**})$ for every Banach space $Y$.
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