Example of infinite dimensional linear spaces where the space is equal to its dual. My understanding is that in finite dimensions, every linear space $V$ is isomorphic to its dual $V^\ast$.
In infinite dimensions, we have that any Hilbert space $\mathcal{H}$ is isomorphic (specifically, anti-isomorphic) to its dual $\mathcal{H}^\ast$ (Riesz Representation Theorem). Furthermore, every Hilbert space is also isomorphic to the square summable sequence space $\ell^2$.
I am wondering if there are examples of infinite dimensional linear spaces where the dual is equal to itself, and the space is not isomorphic to $\ell^2$.
Edit: We assume the underlying field to be $\mathbb{R}$ or $\mathbb{C}$.

If $V$ is a normal infinite dimensional vector space, no additional structures then $$\dim V^*=2^{\dim V}$$ so they can never be isomorphic.
