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Abstract We deal with the approximate solution of initial value problems in infinite-dimensional Banach spaces with a Schauder basis. We only allow finite-dimensional algorithms acting in the spaces R N , with varying N. The error of such algorithms depends on two parameters: the truncation parameters N and a discretization parameter n. For a class of C r right-hand side functions, we define an algorithm with varying N , based on possibly non-uniform mesh, and we analyze its error and cost. For constant N , we show a matching (up to a constant) lower bound on the error of any algorithm in terms of N and n , as N , n → ∞. We stress that in the standard error analysis the dimension N is fixed, and the dependence on N is usually hidden in error coefficient. For a certain model of cost, for many cases of interest, we show tight (up to a constant) upper and lower bounds on the minimal cost of computing an ε -approximation to the solution (the ε -complexity of the problem). The results are illustrated by an example of the initial value problem in the weighted ℓ p space (1 ≤ p < ∞). [ABSTRACT FROM AUTHOR]