Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual basis(for $E^*$). Now we define $\omega:=e^1\wedge\cdots\wedge e^n$ as an element of volume of $E$.
I prove that $g^{\flat}: E\rightarrow E^*$
with rule $(g^{\flat}(u))(v)=g(u,v)$, is an isomorphism ($ \forall u,v\in E$).
Convention: $u\stackrel{g^{\flat}}\mapsto \tilde{u} $ i.e. $\tilde{u}:=g^{\flat}(u)$.
I wish to prove that for any p-form $\theta\in \Lambda^p(E)$, There exist a unique element $\eta\in\Lambda^{(n-p)}(E)$ such that
$\eta(u_1,\cdots, u_{n-p})\omega=\theta\wedge\tilde{u}_1\cdots\wedge \tilde{u}_{n-p}\qquad \forall u_1,\cdots, u_{n-p}\in E$ .
How can I do this?
Of course I guess that should be defined an inner product $g_{\Lambda^k(E)}$ on $\Lambda^k(E)$ for any $0<k<n$ and then use $g^{\flat}_{\Lambda^k(E)}$. what is your method ? and How?
Given a $p$-form $\theta \in \bigwedge^p E^*$, we can define an alternating multilinear map $h \colon E^{n-p} \to \bigwedge^n E$ by $$h(u_1, \ldots, u_{n-p}) = \theta \wedge \tilde{u}_1 \wedge \ldots \wedge \tilde{u}_{n-p}.$$
Let $b \colon \mathbb{R} \to \bigwedge^n E$ be the linear map $$b(t) = t \omega.$$ Because $\bigwedge^n E$ is one-dimensional and $\omega$ is nonzero, this map is an isomorphism. You can think of it as the coordinate system for $\bigwedge^n E$ given by the basis $\{\omega\}$. Observe that, for any $\alpha \in \bigwedge^n E$, the scalar $b^{-1}(\alpha)$ is the unique scalar with the property that $b^{-1}(\alpha)\,\omega = \alpha$.
Composing $h$ with $b^{-1}$ gives a map $H \colon E^{n-p} \to \mathbb{R}$. Looking back at our discussions of $h$ and $b^{-1}$, you can see that $H(u_1, \ldots, u_{n-p})$ is the unique scalar with the property that $$H(u_1, \ldots, u_{n-p})\,\omega = \theta \wedge \tilde{u}_1 \wedge \ldots \wedge \tilde{u}_{n-p}.$$
Since $h$ is multilinear and alternating, $H$ is too. The map $H$ can therefore be expressed as an $(n-p)$-form $\eta$, using the standard correspondence between elements of $\bigwedge^{n-p} E^*$ and alternating multilinear maps $E^{n-p} \to \mathbb{R}$.