Complexification, Realification and Real Structures

The map $b:\mathbb{C}\times V\to V\oplus iV$, $b(z,v)=z(v,0)$, is $\mathbb{R}$-bilinear: it is additive in each variable because the addition in $V\oplus iV$ is componentwise, and for $r\in \mathbb{R}$ one has $b(zr,v)=(zr)(v,0)=z(r(v,0))=z(rv,0)=b(z,rv)$ by [L2].

By [L3] there is a unique $\mathbb{R}$-linear map $\mathrm{\Phi }:\mathbb{C}{\otimes }_{\mathbb{R}}V\to V\oplus iV$ with $\mathrm{\Phi }(z\otimes v)=z(v,0)$.

The map $\mathrm{\Psi }(v+iw)=1\otimes v+i\otimes w$ satisfies $\mathrm{\Psi }\circ \mathrm{\Phi }=\mathrm{id}$ on elementary tensors: writing $z=a+bi$, one has $\mathrm{\Psi }(\mathrm{\Phi }(z\otimes v))=\mathrm{\Psi }(av+ibv)=1\otimes (av)+i\otimes (bv)=a(1\otimes v)+b(i\otimes v)=(a+bi)(1\otimes v)=z\otimes v$ by the scalar action of [L1].

Conversely $\mathrm{\Phi }\circ \mathrm{\Psi }=\mathrm{id}$: $\mathrm{\Phi }(1\otimes v+i\otimes w)=(v,0)+i(w,0)=(v,0)+(0,w)=(v,w)=v+iw$ by [L2].

The map $\mathrm{\Phi }$ is complex-linear: for $z,z^{\mathrm{\prime }}\in \mathbb{C}$, $\mathrm{\Phi }(z^{\mathrm{\prime }}\cdot (z\otimes v))=\mathrm{\Phi }((z^{\mathrm{\prime }}z)\otimes v)=(z^{\mathrm{\prime }}z)(v,0)=z^{\mathrm{\prime }}(z(v,0))=z^{\mathrm{\prime }}\mathrm{\Phi }(z\otimes v)$ by [L1] and [L2], and the identity extends from elementary tensors by additivity.

Steps 3.1 and 3.2 make $\mathrm{\Phi }$ bijective with inverse $\mathrm{\Psi }$, and step 3.3 makes it complex-linear, so $\mathrm{\Phi }$ is the claimed canonical complex-linear isomorphism.

The map $b:\mathbb{C}\times V\to W$ given by $b(z,v)=zf(v)$ is $\mathbb{R}$-bilinear and $\mathbb{R}$-balanced: additivity follows from the linearity of $f$ in $v$ and the distributivity of complex scalar multiplication, and $b(zr,v)=(zr)f(v)=z(rf(v))=zf(rv)=b(z,rv)$ for $r\in \mathbb{R}$.

Uniqueness of any extension: if $G:V_{\mathbb{C}}\to W$ is complex-linear with $G\circ \iota =f$, then $G(z\otimes v)=G(z\cdot (1\otimes v))=zG(1\otimes v)=zf(v)$ on every elementary tensor by [L1], so its values are already determined, and elementary tensors generate $V_{\mathbb{C}}$.

By [L2] there is a unique $\mathbb{R}$-linear map $F:V_{\mathbb{C}}\to W$ with $F(z\otimes v)=zf(v)$.

The map $F$ is complex-linear: by the scalar action of [L1], $F(z^{\mathrm{\prime }}\cdot (z\otimes v))=F((z^{\mathrm{\prime }}z)\otimes v)=(z^{\mathrm{\prime }}z)f(v)=z^{\mathrm{\prime }}(zf(v))=z^{\mathrm{\prime }}F(z\otimes v)$, and additivity extends the identity to all of $V_{\mathbb{C}}$.

The map $F$ satisfies $F\circ \iota =f$ because $F(\iota v)=F(1\otimes v)=f(v)$ by [L1].

Uniqueness: steps 1.2 and 2.1 show that any complex-linear extension $G$ with $G\circ \iota =f$ agrees with $F$ on every elementary tensor, hence everywhere, so $G=F$.

For the uniqueness up to unique isomorphism, apply the universal property of $(V_{\mathbb{C}},\iota )$ to ${\iota }^{\mathrm{\prime }}$ to get a unique complex-linear $u:V_{\mathbb{C}}\to U$ with $u\circ \iota ={\iota }^{\mathrm{\prime }}$, and the universal property of $(U,{\iota }^{\mathrm{\prime }})$ to $\iota$ to get $v:U\to V_{\mathbb{C}}$ with $v\circ {\iota }^{\mathrm{\prime }}=\iota$. Both $v\circ u$ and $\mathrm{id}$ carry $\iota$ to itself, and both $u\circ v$ and $\mathrm{id}$ carry ${\iota }^{\mathrm{\prime }}$ to itself, so step 3.3 forces both composites to be identities; this is the two-application argument recorded in [L3].

Steps 3.1 and 3.2 prove the universal property, step 3.3 its uniqueness clause, and step 4.1 the uniqueness of the representing pair up to unique isomorphism.

By [L2] applied with $f=f^{\mathrm{\prime }}={\mathrm{id}}_{\mathbb{C}}$ and $g={\mathrm{id}}_V$, one has $({\mathrm{id}}_V{)}_{\mathbb{C}}={\mathrm{id}}_{\mathbb{C}}\otimes {\mathrm{id}}_V={\mathrm{id}}_{V_{\mathbb{C}}}$ by [L1].

By [L2] applied with $f=f^{\mathrm{\prime }}={\mathrm{id}}_{\mathbb{C}}$, $g=T$ and $g^{\mathrm{\prime }}=S$, one has $(S\circ T{)}_{\mathbb{C}}={\mathrm{id}}_{\mathbb{C}}\otimes (S\circ T)=({\mathrm{id}}_{\mathbb{C}}\circ {\mathrm{id}}_{\mathbb{C}})\otimes (S\circ T)=({\mathrm{id}}_{\mathbb{C}}\otimes S)\circ ({\mathrm{id}}_{\mathbb{C}}\otimes T)=S_{\mathbb{C}}\circ T_{\mathbb{C}}$ by [L1].

Steps 1.1 and 1.2 are exactly the identity and composition laws of a functor.

In the direct-sum model, $T_{\mathbb{C}}(v+iw)=Tv+iTw$: applying $\mathrm{\Phi }$ to the formula of [L1], $\mathrm{\Phi }(T_{\mathbb{C}}(1\otimes v+i\otimes w))=\mathrm{\Phi }(1\otimes Tv+i\otimes Tw)=Tv+iTw$ by [L2].

If $U\le V$ is a real subspace with real basis $u_1,\dots ,u_m$, then every element of $U\oplus iU$ is ${\sum }_{j\le m}(a_j+i{b}_j)u_j$ with $a_j,b_j\in \mathbb{R}$: the real and imaginary components are real combinations of the $u_j$.

The list $u_1,\dots ,u_m$ is complex-linearly independent in $U\oplus iU$: ${\sum }_j(a_j+i{b}_j)u_j=0$ means $({\sum }_j{a}_j{u}_j,{\sum }_j{b}_j{u}_j)=(0,0)$ in $V\oplus iV$, and the real independence of the $u_j$ forces every $a_j=b_j=0$.

By [L6] and the hypothesis $p\circ i=0$, one has $p_{\mathbb{C}}\circ i_{\mathbb{C}}=0$, hence $\mathrm{im}{i}_{\mathbb{C}}\subseteq \ker p_{\mathbb{C}}$.

$\ker (T_{\mathbb{C}})=\{v+iw:Tv=Tw=0\}=(\ker T)\oplus i(\ker T)$: the equality of the two descriptions is step 1.1, and by [L3] the kernel of the componentwise map is the complexification of $\ker T$.

$\mathrm{im}(T_{\mathbb{C}})=\{Tv+iTw\}=(\mathrm{im}T)\oplus i(\mathrm{im}T)$, again directly from step 1.1.

For a finite-dimensional real subspace $U$, steps 1.2 and 1.3 exhibit $u_1,\dots ,u_m$ as a complex basis of $U\oplus iU$, so ${\dim }_{\mathbb{C}}(U\oplus iU)={\dim }_{\mathbb{R}}U$.

Combining steps 2.1, 2.2 and 2.3 gives ${\mathrm{nullity}}_{\mathbb{C}}(T_{\mathbb{C}})={\dim }_{\mathbb{C}}\ker (T_{\mathbb{C}})={\dim }_{\mathbb{R}}\ker T={\mathrm{nullity}}_{\mathbb{R}}(T)$ and the matching rank identity, with rank and nullity as in [L4].

At the middle term, $\ker p_{\mathbb{C}}=(\ker p)\oplus i(\ker p)=(\mathrm{im}i)\oplus i(\mathrm{im}i)=\mathrm{im}{i}_{\mathbb{C}}$ by [L5] and steps 2.1 and 2.2 applied to $i$ and $p$.

The map $i_{\mathbb{C}}$ is injective because $\ker i_{\mathbb{C}}=(\ker i)\oplus i(\ker i)=0$ by [L3], and $p_{\mathbb{C}}$ is surjective because $\mathrm{im}{p}_{\mathbb{C}}=(\mathrm{im}p)\oplus i(\mathrm{im}p)=W_{\mathbb{C}}$.

By [L5], exactness of the complexified sequence is: $0=\ker i_{\mathbb{C}}$ at $U_{\mathbb{C}}$, $\mathrm{im}{i}_{\mathbb{C}}=\ker p_{\mathbb{C}}$ at $V_{\mathbb{C}}$, and $\mathrm{im}{p}_{\mathbb{C}}=W_{\mathbb{C}}$ at $W_{\mathbb{C}}$; these are step 3.3, step 3.2 and step 3.3 respectively, with the containment of step 1.4 absorbed into the equality.

Steps 2.1 and 2.2 prove the kernel and image formulas, step 3.1 the rank and nullity preservation, and step 4.1 the short-exact-sequence clause.

By [L2], $\{1,i\}$ is an $\mathbb{R}$-basis of $\mathbb{C}$, and by hypothesis $(v_1,\dots ,v_n)$ is an $\mathbb{R}$-basis of $V$; hence [L1] makes $\{1\otimes v_j,i\otimes v_j:j\le n\}$ an $\mathbb{R}$-basis of $\mathbb{C}{\otimes }_{\mathbb{R}}V$.

Transporting through the inverse isomorphism $\mathrm{\Psi }$ of [L3], which sends $1\otimes v_j$ to $v_j$ and $i\otimes v_j$ to $i{v}_j$, the set $\{v_j,i{v}_j:j\le n\}$ is an $\mathbb{R}$-basis of $V\oplus iV$.

The vectors $v_1,\dots ,v_n$ span $V\oplus iV$ over $\mathbb{C}$: every element $v+iw$ is ${\sum }_j{a}_j{v}_j+i{\sum }_j{b}_j{v}_j={\sum }_j(a_j+i{b}_j)v_j$ by the real basis expansion of step 1.2.

The vectors $v_1,\dots ,v_n$ are complex-linearly independent: if ${\sum }_j(a_j+i{b}_j)v_j=0$, then $({\sum }_j{a}_j{v}_j,{\sum }_j{b}_j{v}_j)=(0,0)$, so the real independence in step 1.2 forces every $a_j=b_j=0$.

Steps 2.1 and 2.2 make $(v_1,\dots ,v_n)$ an ordered complex basis of the direct-sum model, and applying $\mathrm{\Phi }$ of [L3] carries it to the ordered complex basis $(\iota v_1,\dots ,\iota v_n)$ of $V_{\mathbb{C}}$.

Both bases have exactly $n$ elements, so by [L5] the complex dimension of $V_{\mathbb{C}}$ equals the real dimension of $V$.

Choose an ordered complex basis $(e_1,\dots ,e_n)$ of $W$, which exists because $W$ is finite-dimensional over $\mathbb{C}$ by [L4].

The list $(e_1,i{e}_1,\dots ,e_n,i{e}_n)$ spans $W_{\mathbb{R}}$: writing $w={\sum }_j{z}_j{e}_j$ and $z_j=a_j+i{b}_j$ by [L2], one has $w={\sum }_j{a}_j{e}_j+{\sum }_j{b}_j(i{e}_j)$ by the scalar-multiplication restriction of [L1].

The list is linearly independent over $\mathbb{R}$: if ${\sum }_j{a}_j{e}_j+{\sum }_j{b}_j(i{e}_j)=0$ with real coefficients, then ${\sum }_j(a_j+i{b}_j)e_j=0$ in $W$, so the complex independence of $(e_1,\dots ,e_n)$ forces every $a_j+i{b}_j=0$ and hence every $a_j=b_j=0$; in particular the displayed entries are pairwise distinct.

By [L3], steps 2.1 and 2.2 make the displayed list a real basis of $W_{\mathbb{R}}$ with $2n$ entries, so [L4] gives ${\dim }_{\mathbb{R}}{W}_{\mathbb{R}}=2n$.

The set $W^{\sigma }$ is a real subspace: it contains $0$, is closed under addition because $\sigma$ is additive by [L1], and is closed under real scalars because $\sigma (rw)=\bar{r}\sigma w=rw$ for $r\in \mathbb{R}$.

Every element of $\mathbb{C}{\otimes }_{\mathbb{R}}{W}^{\sigma }$ has the form $1\otimes A+i\otimes B$ with $A,B\in W^{\sigma }$: a general finite sum is ${\sum }_j{z}_j\otimes w_j={\sum }_j(a_j+i{b}_j)\otimes w_j={\sum }_j(a_j+i{b}_j)\cdot (1\otimes w_j)=1\otimes ({\sum }_j{a}_j{w}_j)+i\otimes ({\sum }_j{b}_j{w}_j)$ by the scalar action of [L5], and the two coefficient sums lie in the real subspace $W^{\sigma }$.

By [L3], the real-linear inclusion $W^{\sigma }\hookrightarrow W$ extends uniquely to a complex-linear map $\theta :\mathbb{C}{\otimes }_{\mathbb{R}}{W}^{\sigma }\to W$ with $\theta (z\otimes w)=zw$.

Surjectivity: for $v\in W$ set $A=(v+\sigma v)\mathrm{/}2$ and $B=(v-\sigma v)\mathrm{/}(2i)$. By [L1] and [L4], $\sigma A=(v+\sigma v)\mathrm{/}2=A$, and $\sigma B=(\sigma v-v)\mathrm{/}(2\bar{i})=(v-\sigma v)\mathrm{/}(2i)=B$ because $\bar{i}=-i$; hence $A,B\in W^{\sigma }$ by [L2] and $\theta (1\otimes A+i\otimes B)=A+iB=v$.

Injectivity: for $A,B\in W^{\sigma }$ one has $\theta (1\otimes A+i\otimes B)=A+iB$ by step 2.1. If $A+iB=0$, applying $\sigma$ and using [L1] gives $A-iB=0$; subtracting the two identities gives $2iB=0$, hence $B=0$ and then $A=0$, so the tensor $1\otimes A+i\otimes B$ is zero.

Steps 2.1, 3.1 and 3.2 make $\theta$ a complex-linear isomorphism, and its restriction to $\iota W^{\sigma }$ is the inclusion because $\theta (1\otimes A)=A$.

For a conjugation $\sigma$, the subspace $W^{\sigma }$ is a real form of $W$: by [L1] the complexification of $W^{\sigma }$ recovers $W$ through the multiplication map, which is exactly the defining condition.

The fixed points of ${\sigma }_{\mathrm{can}}$ on $V_{\mathbb{C}}$ are the embedded copy of $V$: by [L4] an element is uniquely $1\otimes A+i\otimes B$, and ${\sigma }_{\mathrm{can}}(1\otimes A+i\otimes B)=1\otimes A-i\otimes B$ by [L2], which equals itself exactly when $i\otimes B=-i\otimes B$, hence $B=0$.

For a real form $V$ with isomorphism $\theta :V_{\mathbb{C}}\to W$, define ${\sigma }_V(w)=\theta ({\sigma }_{\mathrm{can}}({\theta }^{-1}w))$. It is additive and conjugate-linear because $\theta$ is complex-linear and ${\sigma }_{\mathrm{can}}$ has these properties by [L2], and it is an involution because ${\sigma }_{\mathrm{can}}^{}=\mathrm{id}$.

If $V=W^{\sigma }$ came from a conjugation $\sigma$, then ${\sigma }_V=\sigma$: for $w=A+iB$ with $A,B\in W^{\sigma }$, step 1.3 gives ${\sigma }_V(w)=\theta (1\otimes A-i\otimes B)=A-iB$, while $\sigma (A+iB)=A+\bar{i}B=A-iB$ by [L2].

If ${\sigma }_V$ came from a real form $V$, then $W^{{\sigma }_V}=\theta (\{1\otimes A:A\in V\})=V$: the fixed points of ${\sigma }_V$ in $W$ are the $\theta$-images of the fixed points of ${\sigma }_{\mathrm{can}}$, which step 1.2 identifies with the embedded copy of $V$, and $\theta (1\otimes A)=A$ by the real-form condition.

Steps 1.1, 2.1 and 2.2 show that the two assignments compose to the identity in both orders, so they are inverse bijections.