Give an example of a discontinuous bilinear form. I'm having a hard time with this, I asked a tutor and he couldn't think of an example either.
The bilinear form on a vector space $V$ is a map $f:  V \times V \rightarrow F$ that satisfies linearity in both variables.

Alternatively, if you let $P$ be the space of real-valued polynomials of a real variable s.t. $\int_0^1 | p | < \infty$ for all $p \in P$, then the map $P \times P \to \mathbb{R}$ given by $(p,q) \mapsto \int_0^1 pq$ is bilinear, but discontinuous. In particular, if we choose $p_n = q_n = \sqrt n x^n$, then $p_n \to 0$ but $(p_n, p_n) \mapsto 1/2$ in limit.
You could be more lenient, and choose $V$ just to be the space of real valued continuous functions whose L1 integral is finite, and consider the same map $V \times V \to \mathbb{R}$. It's easy to see that this one isn't continuous, as now you can choose bump-style functions that converge non-uniformly to 0 everywhere, but whose product doesn't.
Again, these are both infinite dimensional.
