Definition 10.2.1 (Complexification).label Let $E$ be a vector space over $\real$, then there exists a pair $(\complex(E), \iota)$ such that:

1. (1)

   $\complex(E)$ is a vector space over $\complex$.

2. (2)

   $\iota: E \to \complex(E)$ is a $\real$-linear map.

3. (U)

   For any pair $(F, T)$ satisfying (1) and (2), there exists a unique $\complex$-linear map $\complex(T): \complex(E) \to F$ such that the following diagram commutes:

   \[\xymatrix{ \mathbb{C}(E) \ar@{->}[r]^{\mathbb{C}(T)} & F \\ E \ar@{->}[u]^{\iota} \ar@{->}[ru]_{T} & }\]

   
   
4. (4)

   $\complex(E) = \iota(E) \oplus i\iota(E)$ as a vector space over $\real$. For each $z \in \complex(E)$ with $z = x + iy$, $x = \text{Re}(x)$ and $y = \text{Im}(y)$ are the real and imaginary parts of $z$.

The pair $(\complex(E), \iota)$ is the complexification of $E$, and

1. (F)

   For any vector space $F$ over $\real$ and $\real$-linear map $T: E \to F$, there exists a unique $\complex$-linear map $\complex(T): \complex(E) \to \complex(F)$ such that the following diagram commutes:

   \[\xymatrix{ \mathbb{C}(E) \ar@{->}[r]^{\mathbb{C}(T)} & \mathbb{C}(F) \\ E \ar@{->}[u]^{\iota} \ar@{->}[r]_{T} & F \ar@{->}[u]_{\iota} }\]

   
   

   which is given by

   \[\complex(T)(x + iy) = Tx + iTy\]