Suppose that we start with $c^0$ and keep taking duals,
$$c^0 \to \ell^1 \to \ell^\infty \to \ell^{\infty*} \to \ell^{\infty **} \to \ell^{\infty***} \to \ell^{\infty****}\to \dots$$
Is this an infinite sequence of distinct spaces? If not, when does it "stabilise", i.e. stop generating new spaces? What about other Lebesgue spaces
$$L^1(\Omega,\mu) \to L^\infty(\Omega,\mu) \to L^{\infty*}(\Omega,\mu) \to L^{\infty **}(\Omega,\mu) \to L^{\infty***}(\Omega,\mu) \to L^{\infty****}(\Omega,\mu)\to \dots?$$
I know that a Banach space is non-reflexive iff its dual is non-reflexive, but do not know anything beyond that. Thanks!
Edit:
Thanks Jose for posting the mathoverflow link. Follow-up questions:
What is the characterisation of $(L^{\infty})^{n*}, n\in\mathbb{N}$? What uses do these spaces have?
Thanks!