How do I change the `enumerate` list format to use letters instead of the default Arabic numerals?

Without any package you could do it by redefining the command \theenumi for formatting the enumi counter. (Also enumii, etc., for nested lists.)

\renewcommand{\theenumi}{\Alph{enumi}}

inside the environment.... Or better, you could use a package like enumitem which allows, e.g.,

\usepackage{enumitem}
...
\begin{enumerate}[label=\Alph*]
\item this is item a
\item another item
\end{enumerate}

Use \alph for lowercase letters, \Alph for uppercase, etc. See the package documentation for more info.

Use the package enumitem.

\usepackage[shortlabels]{enumitem}
.
.
.
\begin{enumerate}[(a)] % (a), (b), (c), ...
\item
\end{enumerate}
.
.
.
\begin{enumerate}[a)] % a), b), c), ...
\item
\end{enumerate}

With enumitem package, we can do as follow:

Preamble:

\usepackage{enumitem}
\newcommand{\subscript}[2]{$#1 _ #2$}

In document use:

\begin{enumerate}[label=(\subscript{E}{{\arabic*}})]
    \item
    Generated by the $f*\tilde{g}$, where $f\in C_c(G)$, $g\in C_c(G)$;
    \item
    Generated by the $h*\tilde{h}$, where $h\in C_c(G)$;
\end{enumerate}