= Codimension of a vector subspace
{title2=$\operatorname{codim}_XW=\dim(X/W)$}
= Vector-space codimension
{synonym}
For a <vector subspace> $W$ of $X$, its codimension is the <dimension> of the <quotient vector space> $X/W$. This definition also applies when $X$ is infinite-dimensional. If $X=W\oplus E$ is a <direct sum>, the map $e\mapsto e+W$ is a <linear isomorphism> from $E$ to $X/W$, so $\operatorname{codim}_XW=\dim E$. In particular, the <kernel> of a nonzero <linear functional> has codimension one: normalize $Te=1$ and write $x=(x-(Tx)e)+(Tx)e$.
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