Complex conjugate of a vector space

If V is a complex vector space, then any two vector spaces with a bijective antilinear map from V to it are all isomorphic. Pick a representative and call it the complex conjugate vector space of V, V*. This comes with a canonical bijective antilinear map from V to V*, called *. The conjugate of the conjugate of V is isomorphic to V. So, we can tweak the definition of the conjugate so that the conjugate of the conjugate of V is none other than V itself and x**=x for all x in V.

Given a linear map $f:V\rightarrow W$, the conjugate linear map $f^{*}:V^{*}\rightarrow W^{*}$ is defined as follows:

$f^{*}(x*)=f(x)^{*}.$

As you may verify for yourself, f* is a linear map.