On isometric embeddings into the set of strongly norm-attaining Lipschitz functions

2022/08/04 by Sheldon Dantas, Dantas, Sheldon, Rubén Medina +5
Mathematics · Computer Science · #Advanced Banach Space Theory #Optimization and Variational Analysis #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.2208.02916

Abstract

In this paper, we provide an infinite metric space M such that the set SNA(M) of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to c₀. This answers a question posed by Antonio Avilés, Gonzalo Martínez Cervantes, Abraham Rueda Zoca, and Pedro Tradacete. On the other hand, we prove that SNA(M) contains an isometric copy of c₀ whenever M is a metric space which is not uniformly discrete. In particular, the latter holds true for infinite compact metric spaces while it does not for proper metric spaces. Some positive results in the non-separable setting are also given.

Related

• Embeddings of infinite-dimensional spaces in the sets of norm-attaining Lipschitz functions
• Infinite dimensional spaces in the set of strongly norm-attaining Lipschitz maps
• Linear structures of norm-attaining Lipschitz functions and their complements
• Closed linear spaces consisting of strongly norm attaining Lipschitz mappings
• On coarse Lipschitz embeddability into c₀(κ)