Examples of when the Riesz representation theorem doesn't hold

Solution:

Consider $c_{00}$, the space of all finitely supported sequences, equipped with the $\|\cdot\|_2$ norm.

The functional $$(x_n)_n \mapsto \sum_{n=1}^\infty \frac{x_n}{n}$$

is bounded on $c_{00}$ but it is not represented by any vector from $c_{00}$.

Namely, it is represented by $\left(\frac1n\right)_n \in \ell^2$, which is in the completion of $c_{00}$.

All examples will be of this form, i.e. the functional will be representable by some vector from the completion of your incomplete inner product space.