Let $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. Let $(X, \| \cdot \|)$ a normed vector space. Let $X*$ be the dual space with the topology $\omega^*$, where $X^*$ is the set of linear and bounded functions $F: X \to \mathbb{F}$ and $\omega^*$ is the smallest topology such that every function of the family $\{f: X^* \to \mathbb{F}\}$ is continuous. Here $f: X^* \to \mathbb{F}$ is defined as $f(\phi) = \phi(x)$, for every $\phi \in X^*$ and every $x \in X$.
Let $+: X^* \times X^* \to X^*$ and $\cdot : \mathbb{F} \times X^* \to X^*$ be both defined as $$+(\phi, \psi) = \phi + \psi, \quad \cdot(c, \phi) = c\phi$$
We also know, that $\beta$ is a basis for $\omega^*$ defined as
$$\beta = \{ \cap_{i = 1}^n f_i^{-1}(V): n \in \mathbb{N}, V \in \tau_{\mathbb{F}}\}$$
that is, $\beta$ is the intersection of all open sets of $\tau_{\mathbb{F}}$ under a finite number of functions $f_i \in \{f: X^* \to \mathbb{F}\}$.
Show that $+$ and $\cdot$ are continuous in the topological space $(X^*, \omega^*)$.
$\textbf{Proof}$.
Let $U \in \omega^*$. Then, $U = \cup \Omega$, for some $\Omega \subseteq \beta$. Then, every element of $\Omega$ is of the form $\cap_{i = 1}^n f_i^{-1}(V)$, for some $n \in \mathbb{N}$ and $f_i \in \{f: X^* \to \mathbb{F}\}$. (We want to prove that $+^{-1}(U) \in \omega^* \times \omega^*$).
If $\Omega = \emptyset$, $+^{-1}(U) = \emptyset \in \omega^* \times \omega^*$. So suppose $\Omega \neq \emptyset$. We have that $$+^{-1}(U) = +^{-1}(\cup \Omega) = \cup_{B \in \Omega} (+^{-1}(B)) = \cup_{B \in \Omega} (+^{-1}(\cap_{i = 1}^{n_B} f_{i_B}^{-1}(V))) = \cup_{B \in \Omega} (\cap_{i = 1}^{n_B} +^{-1}(f_{i_B}^{-1}(V))), \quad \forall V \in \tau_{\mathbb{F}}$$
By definition of $\omega^*$, $f_{i_B}^{-1}(V) \in \omega^*$. Therefore, it is enough to prove that $$+^{-1}(f^{-1}(V)) \in \omega^* \times \omega^*, \quad \forall f \in \{f: X^* \to \mathbb{F}\}, \forall V \in \tau_{\mathbb{F}}$$
We presume that $\beta \times \beta$ is a basis for $\omega^* \times \omega^*$. My intuition says that $+^{-1}(f^{-1}(V)) = (B_1, B_2)$, for some $(B_1, B_2) \in \beta \times \beta$. Since $f(\phi) = \phi(x)$, \begin{align*} +^{-1}(f^{-1}(V)) & = \{(\phi, \psi) \in X^* \times X^*: (\phi + \psi) \in f^{-1}(V) \}\\ & = \{(\phi, \psi) \in X^* \times X^*: (\phi + \psi) \in \{g \in X^*: f(g) \in V\} \}\\ & = \{(\phi, \psi) \in X^* \times X^*: f(\phi + \psi) \in V \}\\ & = \{(\phi, \psi) \in X^* \times X^*: (\phi + \psi)(x) \in V \}\\ & = \{(\phi, \psi) \in X^* \times X^*: (\phi(x)+ \psi (x)) \in V \}\\ & = \{(\phi, \psi) \in X^* \times X^*: (f(\phi)+ f(\psi)) \in V \}\\ & = \{(\phi, \psi) \in X^* \times X^*: f(\phi) \in V - f(\psi)\}\\ & = \{(\phi, \psi) \in X^* \times X^*: \phi \in B_1 := f^{-1}(V - f(\psi)) \in \omega^*\}\\ & = \{(\phi, \psi) \in B_1 \times X^*: \phi \in B_1\}\\ \end{align*}
.. and I don't see how to build my basics $B_1, B_2$. Any help is well-received!