Example of infinite dimensional linear spaces where the space is equal to its dual.

Question

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 $\mathbb{C}$.

Answer 1

There's some confusion here:

• In the context of Hilbert spaces, $\mathcal{H}^*$ is not the full (algebraic) dual of $\mathcal H$. It's the topological dual, that is, the space of all continuous linear forms.
• It is not true that every infinite-dimensional Hilbert space is isomorphic to the space $\ell^2$ of square summable sequences. Only those which are separable.

If we are talking only about the algebraic dual, then no infinite-dimension vector space $V$ is isomorphic to $V^*$.

Answer 2

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