A basic doubt on a problem on direct sum of two vector spaces Let $V$ be a vector space over $\Bbb R$ of dimension $k \lt \infty$. Let $\{e_1, e_2,\dots, e_k\}$ be a basis for $V$. Define
$$ V_{\Bbb C}=V \bigoplus iV=\{u+iv: u,v \in V\}$$
with the addition and sclar multiplication . I am confused about the notation $u+iv$. What is this $+$ ? Once I identify $V$ and $iV$ as subspace of some vector space then only I can add them. Also, what is meant by $iv$? As $V$ is a vector space over $R$ what is the meaning of $iv$ ?  Am I correct ? If it were the following then I don't have any problem :
$$ V_{\Bbb C}=V \bigoplus iV=\{(u,iv): u,v \in V\}$$
 A: I think you'll find it less confusing to use slightly different notation for your complexification.
First, recall that you can construct $\mathbb{C}$ as $\mathbb{R}^2$ together with the multiplication
$$
 (a,b)(c,d) = (ac-bd,ad+bc).
$$
The multiplicative identity then is $1 = (1,0)$, and you have $i = (0,1)$, so that if you identify $\mathbb{R}$ with the subfield $\mathbb{R} \oplus \{0\}$ of $\mathbb{C}$, you get that
$$
  \forall (a,b) \in \mathbb{C}, \quad (a,b) = (a,0) + (b,0)i = a + bi,
$$
as you expect.
Now, suppose you have a real vector space $V$. Then you construct $V_{\mathbb{C}}$ as $V \oplus V$ together with the complex scalar multiplication
$$
 (a,b)(v,w) := (av-bw,aw+bv),
$$
by exact analogy with the construction of $\mathbb{C}$ from $\mathbb{R}^2$. Hence, in particular,
$$
 i(v,w) = (-w,v),
$$
so that if you identify $V$ with the real subspace $V \oplus \{0\}$ of $V_{\mathbb{C}}$, then
$$
 \forall (v,w) \in V_{\mathbb{C}}, \quad (v,w) = (v,0) + i(w,0) = v+iw,
$$
as you wanted.
From here on out, the only issue is what Adam Saltz mentioned in his comment: once you've constructed $V_{\mathbb{C}}$ as above, and identified $V$ with $V \oplus \{0\} \subset V \oplus V =: V_{\mathbb{C}}$, then you can also write
$$
 V_{\mathbb{C}} = V \oplus iV,
$$
where $V \oplus iV$ now denotes an internal direct sum of $V \oplus \{0\}$, identified with $V$, and $\{0\} \oplus V = i(V \oplus \{0\})$, identified with $iV$, within $V_{\mathbb{C}}$.
