This paper follows the generalisation of the classical theory of Diophantine approximation to subspaces of R~n established by W. M. Schmidt in 1967. Let A and B be two subspaces of R~n of respective dimensions d and e with d + e ≤ n. The proximity between A and B is measured by t = min(d, e) canonical angles 0 ≤ θ_1 ≤ · · · ≤ θ_t ≤ π/2 ; we set ψ_j (A, B) = sin θ_j . If B is a rational subspace, its complexity is measured by its height H(B) = covol(B ∩ Z~n). We denote by μ_n(A|e)_j the exponent of approximation defined as the upper bound (possibly equal to +∞) of the set of β > 0 such that the inequality ψj (A, B) ≤ H(B)~(?β) holds for infinitely many rational subspaces B of dimension e. We are interested in the minimal value μ_n(d |e)_j taken by μ_n(A|e)_j when A ranges through the set of subspaces of dimension d of R~n such that for all rational subspaces B of dimension e one has dim(A ∩ B) < j. We show that μ_4(2|2)_1 = 3, μ_5(3|2)_1 ≤ 6 and μ_(2d) (d |?)_1 ≤ 2d~2/(2d ??). We also prove a lower bound in the general case, which implies that μ_n(d |d)_d →1/d as n→+∞.
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