# dg.differential geometry – Does the continuous mapping space between topological manifolds always admit a Banach manifold structure?

Let $$M$$ and $$N$$ be smooth, i.e. $$C^infty$$, manifolds. Suppose that $$M$$ is compact. Then for every $$k geq 0$$ it is well known that $$C^k(M,N)$$ admits the structure of a smooth Banach manifold. I am interested in the continuous case, i.e. $$k = 0$$. I think that then one can drop the regularity of $$M$$ and $$N$$ and still get a Banach manifold $$C(M,N)$$ as one constructs the charts via an exponential map. My question: Is $$C(M,N)$$ a Banach manifold also in the case where $$M$$ and $$N$$ are merely topological manifolds, i.e. $$C^0$$ manifolds? I mean if $$M$$ and $$N$$ admit a $$C^1$$ structure or equivalently a $$C^infty$$ structure, then the statement is obvious, but this is not always the case.