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

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 $V^*$.

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