= Finite codimension in a Banach space
{title2=$\operatorname{codim}_X Y=\dim(X/Y)<\infty$}
= Finite codimension
{synonym}
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.
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