Generalized monotone left-inverses in infinite-dimensional Hilbert spaces

Determine whether every strictly monotone mapping from a closed, convex subset C of an infinite-dimensional Hilbert space H into H admits an extended monotone left-inverse defined on the convex hull of its image.

Background

The main existence and continuity results establish an extended monotone left-inverse for strictly increasing mappings on closed convex subsets of finite-dimensional Euclidean spaces. The proofs rely on finite-dimensional tools, including Helly's theorem and compactness arguments.

The authors observe that the definition of strict monotonicity extends naturally to arbitrary inner product spaces and explicitly ask whether the existence result persists for strictly monotone mappings between closed convex subsets of an infinite-dimensional Hilbert space.