Condition for continuity of bilinear form

Solution 1:

Assume $a$ is continuous at the origint. Since $a(0,0)=0$, by definition of continuity, there exists some $\delta>0$ such that $\left|a(u,v)\right|<1$ for any $\|u\|,\|v\|\leq\delta$ (here I assume maximum norm). Thus, for any $u,v$ we have $$ \left|a(u,v)\right|= \left|a\left(\frac{\|u\|}{\delta}\,\frac{\delta u}{\|u\|}, \frac{\|v\|}{\delta}\,\frac{\delta v}{\|v\|}\right)\right|= \delta^{-2}\|u\|\|v\| \left|a\left(\frac{\delta u}{\|u\|}, \frac{\delta v}{\|v\|}\right)\right|<\delta^{-2}\|u\|\|v\| $$ since $\delta\,u/\|u\|$, $\delta\, v/\|v\| \leq \delta$. Hence, $a$ is bounded.