ok, first of all i'm not sure what you mean by the dual map but i can guess and it looks to me that it should be what you meant:
the dual of a linear map is defined by for all
if that's the definition, then the answer to your question is "yes". to prove this, we only need to show that is onto because, since is an isomorphism, we have
so suppose it is not. then there exists a subspace of such that
now let and define by where note that and is well-defined because the sum of and is direct. clearly for all i.e. contradicting our hypothesis that is an isomorphism.
the coverse is also true, i.e. if is an isomorphism, then is an isomorphism too.