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.

In order to have such an example, you need $V$ to be infinite-dimensional. Here’s a concrete example.
Let $F=\mathbb{R}$, and let $V$ be the set of infinite sequences of real numbers that are eventually $0$ (i.e., that have only finitely many non-zero terms) with the uniform (or sup) norm. $V$ is a subspace of $\ell^\infty$, and in fact of $c_0$. Now define $$f:V\times V\to\mathbb{R}:\langle x,y\rangle\mapsto\sum_{n\in\mathbb{N}}2^nx(n)y(n)\;;$$ for each $\langle x,y\rangle\in V\times V$ only finitely many terms of the sum are non-zero, so $f$ is well-defined, and it’s obviously bilinear.
For $n\in\mathbb{N}$ define $x_n\in V$ by $$x_n(k)=\begin{cases}2^{-n},&\text{if }k=n\\0,&\text{if }k\ne n\end{cases}\;,$$ and let $z$ be the zero vector. Clearly $$\Big\langle\langle x_n,z\rangle:n\in\mathbb{N}\Big\rangle\to\langle z,z\rangle$$ in $V\times V$, since $\|x_n\|_\infty=2^{-n}$, but $f\big(\langle x_n,z\rangle\big)=1$ for every $n\in\mathbb{N}$.
