We obtain several Banach-Stone type theorems for vector-valued functions in this paper. Let X,Y be realcompact or metric spaces, E,F locally convex spaces, and φ a bijective linear map from C(X,E) onto C(Y, F ). If φ preserves zero set containments, i.e., z(f) ⊆ z(g)⇐⇒ z(φ(f)) ⊆ z(φ(g)), ∀ f, g ∈ C(X,E), then X is homeomorphic to Y , and φ is a weighted composition operator. The above conclusio...

We introduce the notion of λ-spaces which is much weaker than cone metric spaces defined by Huang and X. Zhang 2007 . We establish some critical point theorems in the setting of λ-spaces and, in particular, in the setting of complete cone metric spaces. Our results generalize the critical point theorem proposed by Dancs et al. 1983 and the results given by Khanh and Quy 2010 to λ-spaces and con...

We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on the persistence diagrams of suitable filtrations built on top of these spaces. We prove the stability of our signatures under Gromov-Hausdorff perturbations of the spaces. We also extend these results to metric spaces equipped with measures. Our signatures are well-suited for the ...