Is it true that if $W$ is a submodule of $V$ then by duality $W^*$ is a sub-co-module of $V^*$? Can anybody confirm that or give an example when this doesn't happen?
I think it might be necessary to the spaces to be semisimple in order to have correspondence, but I didn't really get why.
Finally, would it change something if $W$ was a co-module of $V$ instead of a module?
Thanks in advance!