For a two-side quaternion normed space $X$, we prove that the left dual space $X'$, and similarly the second left dual space $X''$, are two-side quaternion Banach spaces. The corresponding property for the left quaternion normed spaces fails. Using a nonstandard construction, we succeed to embed the space $X$ two-linearly and isometrically into the second dual space $X''$. Consequently, the notion of reflexivity can be introduced in a natural way in such spaces.