Let $p=p_1^{b_1}...(x-\lambda_i)^{b_i}...p_m^{b_m}$ be the minimal polynomial for an operator $T:V\to V$ and let $f=f_1^{c_1}...(x-\lambda_i)^{c_i}...f_m^{c_m}$ be the characteristic polynomial. Let $M_i=\{v\in V: (T-\lambda_iI)^mv=0 \ \text{for some} \ m\geq 0\}$.
In Axler's text, he claims that $M_i=null((T-\lambda_i)^{dim \ V})$
But in Hoffman and Kunze, the suggestion seems to be that $M_i=null((T-\lambda_i)^{c_i})$
So I'm a bit confused as to which one it is.
They're the same, so it doesn't matter which definition you use. In fact we should have $M_i = \text{null}((T - \lambda_i)^{b_i})$, which is sharp.