= Separating space of a linear map
{title2=$S(T)=\{y:\exists x_n\to0,\ Tx_n\to y\}$}
For an everywhere-defined <linear map> $T:E\to F$ between <Banach spaces>, its separating space measures the possible limiting defect in continuity at zero. It is a <closed linear subspace>: linear combinations combine witnessing sequences, and diagonal choices with $\|x_j\|<1/j$ and $\|Tx_j-y_j\|<1/j$ handle a convergent sequence $y_j$ of defects. The <closed graph theorem> gives $T$ continuous exactly when $S(T)=\{0\}$, since a limit $(x_n,Tx_n)\to(x,y)$ has $y-Tx\in S(T)$.
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