Norm-convergent martingales on tensor products of Banach spaces are considered in a measure-free setting. As a consequence, we obtain the following characterization for convergent martingales on vectorvalued L p -spaces: Let (Ω, Σ, μ) be a probability space, X a Banach space and (Σ n ) an increasing sequence of sub σ -algebras of Σ. In order for (f n , Σ n ) ∞ n=1 to be a convergent martingale in L p (μ, X) (1 p < ∞) it is necessary and sufficient that, for each i ∈ N, there exists a convergent martingale (x (n) i , Σ n ) ∞ n=1 in L p (μ) and y i ∈ X such that, for each n ∈ N, we have (n) i | L p (μ) < ∞ and lim i→∞ y i → 0.