 Separating space of a linear map

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 $\Vert x_j\Vert <1/j$ and $\Vert T{x}_j-y_j\Vert <1/j$ handle a convergent sequence $y_j$ of defects. The closed graph theorem gives $T$ continuous exactly when $S(T)=\{0\}$, since a limit $(x_n,T{x}_n)\to (x,y)$ has $y-Tx\in S(T)$.