A \(Q^ \infty\)-manifold topology of the space of Lipschitz maps (Q1313920)

Consider a nondiscrete metric compactum \(X\) and a separable locally compact ANLE (absolute neighborhood LIP extensor) \(Y\), without isolated points. Denote by LIP\((X,Y)\) the space of Lipschitz maps from \(X\) to \(Y\) with the sup-metric (defining the compact-open topology), \(k\)-LIP\((X,Y) = \{f \in \text{LIP}(X,Y)\mid \text{lip }f \leq k\}\), where \(\text{lip }f\) is the Lipschitz constant of \(f\), LIP\((X,Y)W = \text{dir }\lim n\)- \(LIP(X,Y)\), \(Q^ \infty = \text{dir }\lim Q^ n\) -- the direct limit of Hilbert cubes. Theorem. The space LIP\((X,Y)W\) is a \(Q^ \infty\)-manifold if each point of \(Y\) has a neighborhood \(V\) with a map \(\gamma: V\times I\to Y\) and \(k > 1\) such that \(\gamma(y,0) = y\) and \(k^{-1}| t - t'| \leq d(\gamma(y,t),\gamma(y,t')) \leq k| t - t'|\), for each \(y \in V\) and \(t,t'\in I\).