I am trying to learn about complex differential forms and I'm having some problem understanding the equality presented in the title, which I have seen in Volker Scheidemann's book "Introduction to Complex Analysis in Several Variables". I will leave below a screenshot of the statement in the book.
As he says, each $\mu \in \bigwedge^{k} V_{\mathbb{C}}^{*}$ is a real valued alternating multivariable function on $V_{\mathbb{C}}^{k}$, so to me, the equality does not makes sense as an element on the right seems to be a complex valued function right? What am I missing?
I appreciate all the help in advance.
Here Scheidemann writes $\bigwedge^k V_{\mathbb C}^*=\bigwedge^k V^*\oplus i\bigwedge^k V^*.$ Notice that there are no $\mathbb C$ subscripts on the right hand side. One way to think about it is like this: if you fix a basis $\{v_j\}_{j=1}^n$ for $V,$ then $V_\mathbb C$ has $\mathbb R$-basis $\{v_j, iv_j\}_{j=1}^n.$ Then the real dual space is spanned by $\{\varphi_j,\psi_j\}_{j=1}^n$ where $$\varphi_j(v_k)=\delta_{jk}\\ \varphi_j(iv_k)=0\\ \psi(v_k)=0\\ \psi_j(iv_k)=\delta_{jk}$$ and the $\varphi_j$'s and the $\psi_j$'s are real linear.