infinite-dimensional inner product space

Question

I've been asked to get an example of an infinite-dimensional inner product space, so I wrote this

As an example is there anything wrong with what I wrote?

Answer 1

Example: infinite dimensional Hilbert spaces are inner product spaces and they have infinite dimension. $l^2$ is a well known example of infinite dimensional Hilbert space (and actually, every infinite dimensional separable Hilbert space is isometrically isomorphic to $l^2$).

Your example, including the definition of the usual inner product on $l^2$, is correct. There is only a typo: you should write $\overline{\omega}_i$, not only in the "compact" sum notation, but also when you explicitely write the first terms of the series. Moreover, do not stop to $z_n \overline{\omega}_n$: write '$+...$'. Of course, recall that this series is evaluated w.r.t. the norm induced by the inner product.