Remark

— Given a topological vector space F over $\mathbf{R}$, it is not always the case that there exists an automorphism $u$ of F whose square is $-1_F$; for example, it is not possible to define vector space structure relative to $\mathbf{C}$ on a vector space over $\mathbf{R}$ of finite odd dimension.

Let E be a topological vector space on $\mathbf{C}$, and $E_0$ the topological vector space on $\mathbf{R}$ which underlies E. Every linear variety M in E is also a linear variety in $E_0$, but the converse is false. To avoid confusion we say that a linear variety for a vector space structure relative to $\mathbf{C}$ (resp. relative to $\mathbf{R}$) is a complex (resp. real) linear variety. A complex linear variety of finite dimension $n$ (resp. of finite codimension $n$) is a real linear variety of dimension $2n$ (resp. of codimension $2n$). In order that a real vector subspace M of E should also be a complex vector subspace, it is necessary and sufficient that $iM \subset M$.

Recall that, if E and F are two topological vector spaces on $\mathbf{C}$, then a mapping of E in F is called $\mathbf{C}$-linear (resp. $\mathbf{R}$-linear) if it is a linear mapping for the vector space structures of E and of F relative to $\mathbf{C}$ (resp. $\mathbf{R}$); every $\mathbf{C}$-linear mapping is evidently $\mathbf{R}$-linear but the converse is false. We say that a $\mathbf{C}$-linear form on E is a complex linear form and that an $\mathbf{R}$-linear form on E (i.e. a linear form on $E_0$) is a real linear form. If $f$ is a complex linear form on E, it is clear that the real part $g = \Re f$ and the imaginary part $h = \Im f$ of $f$ are real linear forms; further, the relation $f(ix) = if(x)$ implies the identity $h(x) = -g(ix)$; in other words we have

Conversely, if $g$ is a real linear form on E, then $f(x) = g(x) - ig(ix)$ is the unique complex linear form on E such that $\Re f = g$; and $f$ is continuous if, and only if, $g$ is continuous.

Now let H be a complex hyperplane in E, with the equation $f(x) = \alpha + i\beta$, where $f$ is a complex linear form on E; putting $g = \Re f$, we see that H is the intersection of two real hyperplanes $H_1, H_2$ with equations respectively $g(x) = \alpha$ and $g(ix) = -\beta$; if H is closed, so also are $H_1$ and $H_2$ (I, p. 13, th. 1). Conversely let $H_0$ be a homogeneous real hyperplane, with equation $g(x) = 0$ (where $g$ is a real linear form on E); then H, the intersection of $H_0$ and $iH_0$, is a homogeneous complex hyperplane, and if $f$ is the complex linear form such that $\Re f = g$, then $f(x) = 0$ is the equation of H; if $H_0$ is closed then H also is closed.

Let G be a topological vector space over $\mathbf{R}$ and let $G_{(c)}$ be the vector space on $\mathbf{C}$ obtained from G by extending the field of scalars to $\mathbf{C}$ (A, II, § 5.1). Identify G as a subset of $G_{(c)}$ by the mapping $x \mapsto 1 \otimes x$. The $\mathbf{R}$-linear mapping $(x, y) \mapsto x + i.y$ is then a bijection of $G \times G$ on $G_{(c)}$, by means of which we transfer the product topology of $G \times G$ to $G_{(c)}$. Then $G_{(c)}$ with this topology is a topological vector space on $\mathbf{C}$. We say that $G_{(c)}$ is the complexified topological vector space of G.