Let $U\subset\mathbb{R}^n$ and let $\Omega^p(U)$ denote the vector space of $p$-forms ($p\in\mathbb{N}$). Define the isomorphism $\Phi:\Omega^{1}(U)\to\Omega^{n-1}(U)$ as $$\Phi\left(\omega=\sum_{i=1}^nf_idx_i\right)=\sum_{i=1}^n(-1)^{i-1}f_idx_1\ldots dx_{i-1}dx_{i+1}\ldots dx_n$$ It can be easily shown that $\Phi$ is an isomorphism. Now define for $\omega\in\Omega^{1}(U)$, $$\|\omega\|=\int_U\omega\:\Phi(\omega)$$ We want to show that $\|.\|$ is actually a norm on $\Omega^1(U)$.
So I started by the following. I want to show that $\|\omega\|\geq 0$. Notice that $$\int_U\omega\:\Phi(\omega)=\int_U\sum_{i=1}^n\sum_{j=1}^n(-1)^{j-1}(f_if_j)dx_idx_1\ldots dx_{j-1}dx_{j+1}\ldots dx_n$$ Now notice that the term $(-1)^{j-1}(f_if_j)dx_idx_1\ldots dx_{j-1}dx_{j+1}\ldots dx_n\not=0$ only if $i=j$. If not, there will be two $dx_i$'s and term will be $0$. So the sum can be rewritten $$\int_U\omega\:\Phi(\omega)=\int_U\sum_{i=1}^n(-1)^{i-1}f_i^2dx_idx_1\ldots dx_{i-1}dx_{i+1}\ldots dx_n$$. Now in order to rearrange $dx_idx_1\ldots dx_{i-1}dx_{i+1}\ldots dx_{n}$ to $dx_1\ldots dx_n$ we need to do $i-1$ swaps and we get \begin{align*}\int_U\omega\:\Phi(\omega)&=\int_U\sum_{i=1}^n(-1)^{i-1}\cdot f_i^2\cdot (-1)^{i-1}dx_1\ldots dx_n\\&=\int_U\sum_{i=1}^n f_i^2 dx_1\ldots dx_n\\&=\sum_{i=1}^n\int_Uf_i^2dx_1\ldots dx_n\geq 0\end{align*} Also is easy to show that $\|\omega\|=0$ if and only $\omega=0$.
I got stuck when showing the triangle inequality and scaling. Here my try :to show the triangle inequality, let $\omega,\eta\in\Omega^1(U)$ and using the fact that $\Phi$ is a homomorphism, \begin{align*}\|\omega+\eta\|&=\int_U(\omega+\eta)\Phi(\omega+\eta)=\int_U(\omega+\eta)\left(\Phi(\omega)+\Phi(\eta)\right)\\&=\int_U\omega\Phi(\omega)+\eta\Phi(\omega)+\omega\Phi(\eta)+\eta\Phi(\eta)\\&=\|\omega\|+\|\eta\|+\int_U\eta\Phi(\omega)+\omega\Phi(\eta)\end{align*}
I got suck here.
Can anyone help with the triangle inequality and scaling?
EDIT: If we redefine $\|\omega\|=\sqrt{\int_U\omega\:\Phi(\omega)}$ then $\|\omega\|$ is still greater than 0, equal if and only if $\omega=0$. And for scaling we get $$\|c\omega\|=\sqrt{\int_U c\omega\:\Phi(c\omega)}=\sqrt{c^2\int_U \omega\:\Phi(\omega)}=|c|\sqrt{\int_U \omega\:\Phi(\omega)}=|c|\|\omega\|$$ and what remains to show is the triangle inequality. Any help?