The direct-sum model $V\oplus iV$ of a complexification

Let $V$ be a real vector space. The direct-sum model of the complexification of $V$ is the complex vector space $V\oplus iV$ whose underlying set is $V\times V$, with componentwise addition

$(v,w)+(v^{\mathrm{\prime }},w^{\mathrm{\prime }}):=(v+v^{\mathrm{\prime }},w+w^{\mathrm{\prime }})$

and complex scalar multiplication

$(a+bi)\cdot (v,w):=(av-bw,aw+bv)(a,b\in \mathbb{R},v,w\in V)\mathrm{.}$

An element is written $v+iw$ for the pair $(v,w)$, so that $i(v+iw)=-w+iv$; the distinguished copy of $V$ is the real subspace of pairs $(v,0)$. This is the concrete model compared with the tensor model $V_{\mathbb{C}}$ of Complexification as $\mathbb{C}{\otimes }_{\mathbb{R}}V$ with its canonical real-linear embedding.

The axioms of a complex vector space are checked directly from the displayed formulas; for instance $i(i(v+iw))=i(-w+iv)=-v-iw=-(v+iw)$, matching $i^2=-1$. Writing $v+iw$ is a notation for the pair, not a claim that $iV$ is already a subspace of anything.