Let E be a separable infinite-dimensional Frechet space, and denote
by σ_E the linear involution of E given by x → −x. In joint work with
Jan van Mill, we show that σ_{l^2} is not topologically conjugate to σ_{R^\infty}.
We give topological characterizations of those involutions (order 2 homeomorphisms) of
Frechet spaces with unique fixed points that are topologically conjugate to σ_{l^2}
and to σ_{R^\infty} and apply this to show that σ_E is topologically conjugate to σ_{l^2}
if and only if E contains an infinite dimensional normable subspace. Otherwise,
it is conjugate to σ_{R^\infty}.