  We show symplectic non-squeezing for the KdV equation on the line $\mathbb
R$. This is achieved via finite-dimensional approximation. Our choice of
finite-dimensional Hamiltonian system that effectively approximates the KdV
flow is inspired by the recent breakthrough of Killip and Visan in the
well-posedness theory of KdV in low regularity spaces, relying on its
completely integrable structure. We also prove and exploit a weak
well-posedness result for KdV in $H^{-1}(\mathbb R)$. Furthermore, the
employment of our methods provides a new concise proof for the known result of
symplectic non-squeezing for the same equation on the circle $\mathbb T$, first
proved by Colliander, Keel, Staffilani, Takaoka, and Tao.
