  Let k be an infinite field, I an infinite set, V a k-vector-space, and
g:k^I\to V a k-linear map. It is shown that if dim_k(V) is not too large (under
various hypotheses on card(k) and card(I), if it is finite, respectively
countable, respectively < card(k)), then ker(g) must contain elements
(u_i)_{i\in I} with all but finitely many components u_i nonzero.
  These results are used to prove that any homomorphism from a direct product
\prod_I A_i of not-necessarily-associative algebras A_i onto an algebra B,
where dim_k(B) is not too large (in the same senses) must factor through the
projection of \prod_I A_i onto the product of finitely many of the A_i, modulo
a map into the subalgebra \{b\in B | bB=Bb=\{0\}\}\subseteq B.
  Detailed consequences are noted in the case where the A_i are Lie algebras.
