# 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 $.

**How to Cite This Entry:**

Complexification of a vector space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Complexification_of_a_vector_space&oldid=46433