For $A, B ∈ 𝓑(X)$ and $V, W ∈ 𝒢_k X$ with $d(V, W) < δ$, take a minimiser $x₀$ of $‖A ·‖$ on the compact unit sphere of $V$ and a point $y₀$ of the unit sphere of $W$ within $δ$ of it; then $g_B(W) ≤ g_A(V) + (‖A‖ + ‖B - A‖) δ + ‖B - A‖$. The same bound with the roles exchanged gives continuity.