Discontinuous bilinear form separately continuous

Question

Do you have an example of a real normed space $V$ and a bilinear form $B : V \times V \to \mathbb R$ that is discontinuous but such that $B$ is separately continuous for each variable?

$V$ has to be not complete in order to not conflict Banach-Steinhaus theorem.

Answer 1

Let $c_{00}\subset \ell^\infty$ be the space of sequences that are eventually zero. Define a bilinear form on it by $$B(x,y)=\sum_k x_k y_k$$ For each fixed $x$, the linear functional $y\mapsto B(x,y)$ is bounded: $$\left|\sum_k x_k y_k \right|\le \|y\|_\infty \sum_k |x_k|$$ (And same with $x,y$ interchanged.)

The form $B$ is not jointly continuous: if $x_k=1/\sqrt{n}$ for $k\le n$ and $x_k =0$ otherwise, then the $\ell_\infty$ norm of $x$ can be arbitrarily small, yet $B(x,x)=1$.

Answer 2

Let $V$ be the subspace of $L^1(0,1)$ consisting of polynomials, and $$B(f, g)=∫_0^1 f g$$ the bilinear form $B$ is continuous in each variables but it is not continuous.