  Let $(X_i, \mathcal{F}_i)_{i\geq1}$ be a martingale difference sequence in a
smooth Banach space. Let $S_n=\sum_{i=1}^nX_i, n\geq 1,$ be the partial sums of
$(X_i, \mathcal{F}_i)_{i\geq 1}$. We give upper bounds on the quantity
$\mathbb{P}\left(\max_{1\leq k\leq n}\lVert S_k\rVert>nx\right)$ in terms of $
n\geq 1$ and $x>0$ in two different situations: when the martingale differences
have uniformly bounded exponential moments and when the decay of the tail of
the increments is polynomial.
