More On Embedding an Affine Space in a Vector Space

for all a1 . . . , am ∈ E, all v1, . . . , vm ∈ −→E , and all λ1, . . . , λm ∈ R. Furthermore, for λi = 0, 1 ≤ i ≤ m, we have f̂(v1 +̂ λ1a1, . . . , vm +̂ λmam) = λ1 · · ·λmf(a1 + λ−1 1 v1, . . . , am + λ−1 m vm). Proof . Let us assume that f̂ exists. We first prove by induction on k, 1 ≤ k ≤ m, that f̂(a1, . . . , vi1 , . . . , vik , . . . , am) = fS(vi1 , . . . , vik), for every S ⊆ {1, . . . ,m}, such that S = {i1, . . . , ik} and k = |S|, for all a1 . . . , am ∈ E, and all v1, . . . , vm ∈ −→ E . For k = 1, assuming for simplicity of notation that i1 = 1, for any a1 ∈ E, since f̂ is m-linear, we have f̂(a1 + v1, a2, . . . , am) = f̂(a1, a2, . . . , am) + f̂(v1, a2, . . . , am),