Show that $\mathscr{E}(\Omega)$ is a complete metric space, i.e. the Frechet Space.
So we must show that $\mathscr{E}(\Omega)$ is complete. Thus let ${f_{n}}$ be a Cauchy Sequence in $\mathscr{E}(\Omega)$, we must show that ${f_{n}}$ converges to some $f$ in $\mathscr{E}(\Omega)$.
Since ${f_{n}}$ is a Cauchy sequence then it must bounded, i.e. there exists and $M$ such that $|{f_{n}}| \leq M$ for all $n\geq \mathbb{N}$. Set $Q = \{f \in \mathscr{E}(\Omega) \mid \text{there exists infinitely many $n$ such that $f_{n} \geq f$} \}$. Indeed $M\in Q$ and M is an upperbound of Q. That is, $Q$ is a nonempty subset of $\mathscr{E}(\Omega)$ that is bounded above. Hence,
$\beta =$ sup $Q$ exists, i.e. $\beta \in \mathscr{E}(\Omega)$
From here I go on to show that $\beta + \epsilon/2 \notin Q$ and $f_{n} < \beta + \epsilon/2$ for infinitely many n. Then I fix one such index, say $n_0 \geq N$ such that $\beta - \epsilon/2 \leq f \leq f_{n_0} \leq \beta + \epsilon/2$
$\implies$ $|f_{n_0} - \beta| < \epsilon/2$ Thus, for all $n\geq N$
$|f_n - \beta| = |f_n - f_{n_0} + f_{n_0}- \beta| \leq |f_n - f_{n_0}| + |f_{n_0}- \beta| \leq \epsilon/2 + \epsilon/2 = \epsilon$
Hence, $\mathscr{E}(\Omega)$ is a complete metric space. I don't believe this is right since I didn't use the definition for the convergence of the smooth functions.
Define $\mathscr{E}(\Omega)$ as the space of all smooth functions, $\varphi \in C^{\infty}(\Omega)$, equipped with convergence (topology) as follows:
$\varphi_{n} \rightarrow \varphi \text{ in } \mathscr{E}(\Omega) \text{ as } n \rightarrow \infty$ if for every compact K $\subset \Omega$ and multi-index $\alpha$, $\partial_{x}^{\alpha} \varphi_{n} \Rightarrow \partial_{x}^{\alpha} \varphi\text{ uniformly on } K \text{ as } n \rightarrow \infty$.
$\mathscr{E}(\Omega)$ can be equipped with semi-norms and that provide the same convergence. Indeed, define for every given compact $K$ and $l$ the semi-norm
$p_{K,l}(\varphi) = \sum_{|\alpha| \leq l}$ sup$_{K}|D^{\alpha}\varphi|$
Thus we have semi-norms,
$\varphi \rightarrow \sum_{|\alpha| \leq l}$ sup$_{K}|D^{\alpha}\varphi|$
where $K$ ranges over all compact subsets of $\Omega$ and $l$ over all integers $\geq 0$