In my numeric script there is a unproved theorem, saying that a bilinear form $a \colon V\times V \to \mathbb{R}$ on a normed vector space $V$ is continuous if and only if
$$|a(v,w)| \leq c \, \|v\| \, \|w\|$$
holds for all $v,w\in V$ for some $c > 0$.
My first question is: what is ment by a continuous bilinearform? Is it according to the norm $\| (v,w) \| := \max \{ \|v\| , \|w\| \}$ (which is equivalent to $ \|(v,w)\| = \|v\| + \|w\|$) ?
If so, then I agree that such a bilinear form is continuous but I don't see that a continuous bilinear form is bounded as above. Can anyone explain this to me?
Best Answer
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.