As the title says, I'm wondering if involutions in infinite dimensional vector spaces always have the entire space as its eigenspace? This is of course true for finite vector spaces, but I've never seen any proofs treating the infinite case.
Obviously they need to span the entirety of the space, so intuitively I feel like this should be the case, but I'm not sure how one would go about proving it if that is so.
EDIT: Over the reals, specifically.