What I know so far is
- In finite dimensions the dual space is continuous, has the same dimension as the space, and the "dual basis" is in fact a basis.
- the Riesz representation theorem proves the existence of a conjugate-linear isometric bijection between an IPS and its continuous dual.
- a linear bijection between vector spaces is an isomorphism, but does this extend to a conjugate-linear bijection ? (presumably so for a real IPS, but what about a complex IPS).
- the "dual basis" of linear functionals is linearly independent in the dual space and (I think) continuous. so that $dim(V') \ge dim(V)$
- I haven't seen any statement anywhere affirming my question, so I suspect that the answer is no.
So, is an infinite dimensional IPS isomorphic to its continuous dual, and is the "dual basis" a basis for the continuous dual ? If not, is there a simple counterexample ?
Update: I see some reference suggesting that this may be true in a Hilbert space, but not in general (http://www.solitaryroad.com/c855.html)