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We derive exponential bounds on probabilities of large deviations for "lighttail" martingales taking values in finite-dimensional normed spaces. Ourprimary emphasis is on the case where the bounds are dimension-independent ornearly so. We demonstrate that this is the case when the norm on the space canbe approximated, within an absolute constant factor, by a norm which isdifferentiable on the unit sphere with a Lipschitz continuous gradient. We alsopresent various examples of spaces possessing the latter property.

Large Deviations of Vector-valued Martingales in 2-Smooth Normed Spaces