Finite codimension in a Banach space (${\mathrm{codim}}_{X}Y=\dim (X/Y)<\infty$)

For a closed linear subspace $Y$ of a Banach space $X$, finite codimension means the vector quotient $X/Y$ has finite dimension. This definition does not subtract two infinite dimensions. A local stable manifold has the codimension of its tangent subspace. If the linearized phase space has a direct sum decomposition $X=E^s\oplus E^u$ with $\dim E^u=1$, then $X/E^s$ is isomorphic to $E^u$, so a local stable manifold tangent to $E^s$ has codimension one.