Complexification of a vector space

The complex vector space $V ^ {\mathbf C }$ obtained from the real vector space $V$ by extending the field of scalars. The space $V ^ {\mathbf C }$ is defined as the tensor product $V \otimes _ {\mathbf R } \mathbf C$. It can also be defined as the set of formal expressions $x + i y$, where $x , y \in V$, with the operations of addition and multiplication by complex numbers defined in the usual way. The space $V$ is contained in $V ^ {\mathbf C }$ as a real subspace and is called a real form of $V ^ {\mathbf C }$. Every basis of $V$ is a basis of $V ^ {\mathbf C }$( over $\mathbf C$). In particular, $\mathop{\rm dim} _ {\mathbf C } V ^ {\mathbf C } = \mathop{\rm dim} _ {\mathbf R } V$. The operation $V \mapsto V ^ {\mathbf C }$ is a functor from the category of vector spaces over $\mathbf R$ into the category of vector space over $\mathbf C$.