In this article Geometric wave equation by Stefan Waldmann he has at page 7
For a chart $(U, x)$ we consider a compact subset $K \subseteq U$ together with a collection $\left\{e_{\alpha}\right\}_{\alpha=1, \ldots, N}$ of local sections $e_{\alpha} \in \Gamma^{\infty}\left(\left.E\right|_{U}\right)$ such that $\left\{e_{\alpha}(p)\right\}_{\alpha=1, \ldots, N}$ is a basis of the fiber $E_{p}$. We always assume that $U$ is sufficiently small or e.g. contractible such that local base sections exist. The collection $\left\{e_{\alpha}\right\}_{\alpha=1, \ldots, N}$ will also be called a local frame. The dual frame will then be denoted by $\left\{e^{\alpha}\right\}_{\alpha=1, \ldots, N}$ where $e^{\alpha} \in \Gamma^{\infty}\left(\left.E^{*}\right|_{U}\right)$ are the local sections with $e^{\alpha}\left(e_{\beta}\right)=\delta_{\beta}^{\alpha}$. For $s \in \Gamma^{\infty}(E)$ we have unique functions $s^{\alpha}=e^{\alpha}(s) \in \mathcal{C}^{\infty}(U)$ such that $$ \left.s\right|_{U}=s^{\alpha} e_{\alpha} $$
Is $\delta_{\beta}^{\alpha}$ a function or a number? If it is a number how
this operation $e^{\alpha}\left(e_{\beta}\right)$ is defined?
For a finite dimensional vector space $V$ over a field $K$ and a basis $(b_\alpha)$, its dual basis $(b^\alpha)$ is defined by the properties
$b^\alpha(b_\alpha)\ =\ 1$
$b^\alpha(b_\beta)\ =\ 0 \ \text{ if } \alpha\ne \beta\,.$
This can be written by a single equation: $b^\alpha(b_\beta)=\delta^\alpha_\beta$ where $\delta^\alpha_\beta$ is the Kronecker delta, which is a number, it's $1$ if $\alpha=\beta$ and $0$ if $\alpha\ne\beta$.
Note that the elements of the dual space $V^*$ are linear functionals $V\to K$ and as such they are uniquely determined by their values on a given basis.
Also note that the functional $b^\alpha$ just assigns the $\alpha$th coordinate of any vector $\in V$ when written in the basis $(b_\beta)$.