Complex conjugate of a vector space

In mathematics, the complex conjugate of a complex vector space ${\displaystyle V\,}$ is a complex vector space ${\displaystyle {\overline {V}}}$ that has the same elements and additive group structure as ${\displaystyle V,}$ but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of ${\displaystyle {\overline {V}}}$ satisfies

where ${\displaystyle *}$ is the scalar multiplication of ${\displaystyle {\overline {V}}}$ and ${\displaystyle \cdot }$ is the scalar multiplication of ${\displaystyle V.}$ The letter ${\displaystyle v}$ stands for a vector in ${\displaystyle V,}$ ${\displaystyle \alpha }$ is a complex number, and ${\displaystyle {\overline {\alpha }}}$ denotes the complex conjugate of ${\displaystyle \alpha .}$

More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure ${\displaystyle J}$ (different multiplication by ${\displaystyle i}$).