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 $.