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.