Let $X_1, X_2$ be two vector spaces. For each $x_1 \in X_1$, define the operator $\theta_{x_1}\colon X_1 \rightarrow X_1$ by $\theta_{x_1}(x_1') = x_1 + x_1'$. Let $\hat{p}_1 = \{p^1_{\omega_1}\}_{\omega_1 \in \Omega_1}$, $\hat{p}_2 = \{p^1_{\omega_2}\}_{\omega_2 \in \Omega_2}$ be two families of seminorms in $X_1$ and $X_2$, respectively. Let $f\colon X_1 \rightarrow X_2$. Then $f$ is continuous in $x_1 \in X_1$ if and only if for every $\omega_2 \in \Omega_2$ and every $\epsilon > 0$, there is a finite subset $\Omega_{x_1, \omega_2, \epsilon} \subset \Omega_1$ and $\delta_{x_1, \omega_2, \epsilon} > 0$ such that for every $x_1' \in X_1$ such that $p^1_{\omega_1}(x_1' - x_1) \leq \delta_{x_1, \omega_2, \epsilon}$, $\omega_1 \in \Omega_{x_1, \omega_2, \epsilon}$, we have that $p^2_{\omega_2}(f(x_1') - f(x_1)) < \epsilon$.
I am really stucked in this question. I don't know what to do about this finite subset...