This question seems to me quite basic, but I cannot find the answer.
In order to define coaction of dual Steenrod algebra, we take action on cohomology (here all homology and cohomology are taken with coefficients in a field):
$A\otimes H^*X\rightarrow H^*X$
and dualize it, so we get
$(H^*X)^*\rightarrow A^*\otimes (H^*X)^*$.
Here comes the question: how do I know that $(H^*X)^*\approx H_*X$? Answer should be, since we are in category of vector spaces, that "we take a evaluation homomorphism into bi-dual space". However, the evaluation homomorphism is iso iff the input vector space was finite dimensional. So if I have a space such that its homology is of finite type, that is ok. But this coaction is defined for every space. How?