Lipschitz condition in infinite dimensional vector spaces

Question

If we have that $T:V \times W \rightarrow Y$ multilinear and $V,W$ are infinite-dimensional normed vector spaces.(the finite-dimensional proof is easy, since you can use compactness of the boundary of the unite sphere, but this one seems a little bit tougher). Is it then true that if $\|T(x,y)\|\le C(x) \|y\|$ and $\|T(x,y)\|\le D(y)\|x\|$,there exists a constant $K$ such that $\|T(x,y)\|\le K \|x|\| \|y\|$

Answer 1

In general, separate continuity does not imply joint continuity (unless $X$ or $Y$ is complete, as pointed out by Daniel).

Here is one example. Let $X=Y=c_{00}$, the space of all sequences $(x_n)\in\mathbb R^{\mathbb N}$ which are eventually $0$ (i.e. all but finitely many $x_n$ are $0$). Let the norm on $c_{00}$ be $\Vert\,\cdot\,\Vert_\infty$. Define a bilinear functional $B:c_{00}\times c_{00}\to \mathbb R$ as follows: $$B((x_n),(y_n))=\sum_{n\in\mathbb N} x_n y_n\, . $$ This makes sense because the sum is in fact a finite sum for any given pair $((x_n),(y_n))$.

It is straightforward to check that $B$ is separately continuous. On the other hand, $B$ is not continuous: if you take any sequence of positive numbers $(a_n)$ such that $a_n\to 0$ and $\sum_0^\infty a_n=\infty$ and if you define, for each $N\in\mathbb N$, the two following elements of $c_{00}$: $$u^N=(a_1,\dots ,a_N,0, 0\dots,)\, , $$ $$ e^N=(1,\dots ,1,0,0\dots )\, ,$$ then the two sequences $(u^N)$ and $(e^N)$ are bounded in the space $c_{00}$, but $$B(u^N,e^N)=\sum_{n=0}^N a_n\to\infty\, .$$

One could suspect that whenever you take two non-complete normed spaces $X$, $Y$, it should be possible to find a bilinear functional on $X\times Y$ which is separately continuous but not continuous. This is, however, not true. In fact, there exist non-complete normed spaces which are nevertheless Baire spaces; see e.g. http://www.mat.unimi.it/users/libor/AnConvessa/Baire_incompleto.pdf. In such spaces, the uniform boundedness principle is availabe, and hence separate continuity implies joint continuity for bilinear maps.