Let k be a field, G be an abelian group and $$r\in \mathbb N.$$ r∈N. Let L be an infinite dimensional k-vector space. For any $$m\in {\text {End}}_k(L)$$ m∈Endk(L) we denote by $$r(m)\in [0,\infty ]$$ r(m)∈[0,∞] the rank of m. We define by $$R(G,r,k)\in [0,\infty ]$$ R(G,r,k)∈[0,∞] the minimal R such that for any map $$A:G \rightarrow {\text {End}}_k(L)$$ A:G→Endk(L) with $$r(A(g'+g'')-A(g')-A(g''))\le r$$ r(A(g′+g′′)-A(g′)-A(g′′))≤r , $$g',g''\in G$$ g′,g′′∈G there exists a homomorphism $$\chi :G\rightarrow {\text {End}}_k(L)$$ χ:G→Endk(L) such that $$r(A(g)-\chi (g))\le R(G, r, k)$$ r(A(g)-χ(g))≤R(G,r,k) for all $$g\in G$$ g∈G . We show the finiteness of R(G, r, k) for the case when k is a finite field, $$G=V$$ G=V is a k-vector space V of countable dimension. We actually prove a generalization of this result. In addition we introduce a notion of Approximate Cohomology groups $$H^k_\mathcal F(V,M)$$ HFk(V,M) [which is a purely algebraic analogue of the notion of $$\epsilon $$ ϵ -representation (Kazhdan in Isr. J. Math. 43:315–323, 1982)] and interperate our result as a computation of the group $$H^1_\mathcal F(V,M)$$ HF1(V,M) for some V-modules M.