Exercise 2.10

Proof. Let $(X_1,X_2,Z)$ be topological spaces and $B$ a bilinear mapping

From now on, $x=(x_1,x_2)$ denotes an arbitrary element of $X_1\times X_2$. We henceforth assume that $B$ is continuous at the origin $(0,0)$ of $X_1\times X_2$, given an arbitrary balanced open subset $W$ of $Z$, there exists in $X_i$ ( $i=1,2$) a balanced open subset $U_i$ such that

In such context, ${\lambda }_i(x)$ is chosen greater than ${\mu }_i(x_i)=\inf <!--\; FUNCTION\; APPLICATION\; -->\{r>0:x_i\in r\cdot U_i\}$; see [1.33] of Functional Analysis for further reading about the Minkowski functionals $\mu$. In other words, $x_i$ lies in ${\lambda }_i(x)U_i$, since $U_i$ is balanced. Thus,

Pick $p=(p_1,p_2)$ in $X_1\times X_2$, and let $q=(q_1,q_2)$ range over $X\times Y$, as a first step: It directly follows from (5 ) that

We now restrict $q$ to a particular neighborhood of $p$. More specifically,

which implies

(the equality at the left is valid, since $U_i=-U_i$). The special case

implies that

since $W$ is balanced. $W$ being arbitrary, we have so established the continuousness of $B$ at arbitrary $p$; which achieves the proof. □