Let $X$ a Hilbert space. I will write BWON for a basic weak open neighbourhood of $0$ in $X$.
Let $U$ a BWON, and for each $1\leq i \leq n$, let $f_i: X \to X$ be a weakly continuous map. I want to show that there exists some other basic weak open neighbourhood $V$ with the property that for each $1\leq i \leq n$
\begin{equation} 2V + 2f_i(V) \subseteq U. \end{equation}
I am not yet fully comfortable with the notion of weak topology, so I would appreciate if you could let me know if the following attempt at a proof works.
Suppose $V_1, \dots ,V_n$ are BWONs with the property that for each $1\leq i \leq n$ \begin{equation} 2V_i + 2f_i(V_i) \subseteq U. \end{equation} Then we may find some BWON $V$ such that \begin{equation} V \subseteq \bigcap_{1\leq i\leq n} V_i. \end{equation} Then $V$ has the required property that for each $1\leq i \leq n$ \begin{equation} 2V + 2f_i(V) \subseteq U. \end{equation} So if we can show that we can find a $V_1$ such that $2V_1 + 2f_1(V_1) \subseteq U$, we are done.
Since $f_1$ is weakly continuous, then so is $g_1 = 2I + 2f_1$. Then since $U$ is open, then $g_1^{-1}(U)$ is a weakly open set containing $0$. Therefore we may find a BWON $V_1$ contained in $g_1^{-1}(U)$, and so
\begin{equation} 2V_1 + 2f_1V_1 = g(V_1) \subseteq g(g_1^{-1}(U))\subseteq U, \end{equation} as required.
I would appreciate if you let me know if this works, and even corrections of small mistakes are welcome.
I think you argument is correct. The only thing I would mention is that $ \bigcap_{1\leq i\leq n} V_i$ is open because it is a finite intersection of open sets.