The Proof details for the first theorem in functional analysis is given in this website:
I have one questions for the proof of lemma 5 given here.
(1) Why this claim hold? from equality $0+0=0$ and continuity of addition in topological vector spaces it follows that there exist neighborhoods of zero $U_1$, $U_2$ such that $U_1+U_2 \subset W$. Given $W$ is an open set including $0$, $f^{-1}(W)$ is an open set $(f:X\times X \longrightarrow X)$, how can we find some open set $U_1$, $U_2$ such that $U_1+U_2 \subset W$.
The base of topology of $X\times X$ is a family of sets $\{U_1\times U_2:U_1\mbox{ open in } X,\;U_2\mbox{ open in } X\}$. Since $f^{-1}(W)$ is open in $X\times X$ it contains some element of the base of the topology of $X\times X$. Hence we have open subsets $U_1$, $U_2$ in $X$ such that $U_1\times U_2\subset f^{-1}(W)$. As the consequence $f(U_1,U_2)\subset W$. In that particulat case $f$ is the addition in $X$, so we conclude $U_1+U_2\subset W$.