TITLE: Examples of when the Riesz representation theorem doesn't hold
QUESTION [6 upvotes]: I was wondering if anyone could give me some interesting "counter examples" to the Riesz representation theorem about functionals over Hilbert spaces.
When I say counter examples, I'm obviously talking about examples where some of the basic assumptions of the theorem aren't met, so the theorem doesn't hold.
In other words - could you show me some non-trivial examples of functionals over inner-product spaces that cannot be expressed as an inner-product with some vector in the vector space?
I already have an example from $C [0, 1]$ based on the standard $L^2$ integral inner-product, but I was wondering if anyone could enlighten me with a more interesting example.
I don't have much background, but I'm very interested to hear about this topic, and I'd appreciate it if you could give full explanations so I could understand.
Thanks in advance

REPLY [4 votes]: 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.