= Injections between successive kernel quotients
{title2=$\ker T^{i+1}/\ker T^i\hookrightarrow\ker T^i/\ker T^{i-1}$}
For a <linear map> $T:V\to V$, set $K_i=\ker T^i$ and $K_0=0$. Since $T(K_{i+1})\subseteq K_i$ and $T(K_i)\subseteq K_{i-1}$, the rule $v+K_i\mapsto Tv+K_{i-1}$ defines a <linear map> $K_{i+1}/K_i\to K_i/K_{i-1}$. It is <injective>: if $Tv\in K_{i-1}$, then $T^iv=0$, so $v\in K_i$. In finite <dimension>, this proves
$$
\dim K_{i+1}-\dim K_i\leq\dim K_i-\dim K_{i-1}.
$$
The <linear map> need not be <surjective>: for $T=0$ on a nonzero <vector space>, its $i=1$ instance is $0\to V$. For a <nilpotent linear map>, these <dimensions> are the increments in the <kernel dimensions of powers of a nilpotent map>.
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