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I have few questions about forms that I am trying to grasp. So, first of all, let us consider the following things according to the order of difficulties.

First, we have space $L_{\mathbb{R}}(\mathbb{C},\mathbb{R})$ which is the space of all linear real forms. We have the following operators $dx : h \mapsto Re(h)$ and $dy : h \mapsto Im(h)$. Let us consider first $L_{\mathbb{R}}(\mathbb{C},\mathbb{R})$ as a real vector space. Suppose we have the operator $\phi \in L_{\mathbb{R}}(\mathbb{C},\mathbb{R})$.

Then $\phi(x) = \phi(x_1 + x_2i) = \phi(dx(x) + i dy(x)) = \phi(dx(x)) + \phi(dy(x)i) = \phi \circ dx + \phi \circ i *dy$. As $\{1,i\}$ forms a real basis for $\mathbb{C}$ we have $\{dx,dy\}$ forms a real basis for $L_{\mathbb{R}}(\mathbb{C},\mathbb{R})$. Is my reasoning correct?

A (1,k) form is a $C^k$ map from $\psi : U \rightarrow L_{\mathbb{R}}(\mathbb{C},\mathbb{R})$ where $U \subset \mathbb{C}$ is open. Any form (1,k) form $\omega$ can be written as:

w = P dx + Q dy, where $Q,P : U \rightarrow \mathbb{C}$.

I don't understand why is this case can someone explain this?

Finally a (2,k) form is defined as $C^k$ map $\phi : U \rightarrow \mathbb{B}(\mathbb{R}^2\times\mathbb{R}^2,\mathbb{C})$ the complex vector space of alternating bilinear mappings.

So I don't understand here what is the topology of $\mathbb{B}(\mathbb{R}^2\times\mathbb{R}^2,\mathbb{C})$ ? I mean I understand the first case the topology of $L_{\mathbb{R}}(\mathbb{C},\mathbb{R})$ is given by the supermum norm which exists as any linear map is continuous (hence bounded). I want a complete understanding of those things. Please be as detailed in your answer as much as possible.

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I am a little bit confused about what are you trying to do. If you just want to define differential forms with complex coefficients then take a look at complexification. On Vladimir Arnold book Ordinary Differential Equations there is a good presentation on complexification.

The idea is to complexify the space of real differential forms. About the topology of the bilinear forms, since this is a vector space of finite dimension, any norm on it generate the same topology. So you can take any norm you want to generate the topology.

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