How can an infinite dimensional vector space like (most of) the Lebesgue or Sobolev spaces even be reflexive? If an infinite dimensional vector space cannot be isomorphic to its dual space, the how can it be isomorphic to the bidual? Would the bidual space not be "even bigger" than the first dual space?
The context of my question is trying to understand why in his 1972 paper on weak symplectic forms, Jerrold Marsden claims that if a manifold is modeled on a reflexive space, then the canonical symplectic form is non degenerate (aka regular or strongly non degenerate). I thought in the infinite dimensional setting it can only be weakly non degenerate as there is no isomorphism between a vector space and its dual.