On the reconstruction of locally convex spaces from their groups of homeomorphisms

Let X and Y be normal locally convex spaces that have a nonempty open set which intersect every
straight line in a bounded set, and let H(X), H(Y ) denote the groups of self-homeomorphisms of X
and Y respectively. Our main goal is to prove the following reconstruction theorem. If there is an isomorphism ϕ between H(X) and H(Y ), then there exists a homeomorphism π between X and Y such that for every h ∈ H(X), ϕ(h) = π ◦ h ◦ π−1