Density of Lipschitz continuous functions

Stream: maths

[Yury G. Kudryashov]

Hi, under what conditions on metric spaces X, Y, Lipschitz continuous functions are dense in C(X, Y)? E.g., is it true if both X and Y are closed balls in Banach spaces?

[Jireh Loreaux]

This paper claims to provide results concerning when uniformly continuous functions between Banach spaces can be approximated by Lipschitz functions.

[Yury G. Kudryashov]

I guess, this means that the answer is "generally, no", and we probably need Schauder fixed point theorem to get Peano exisence theorem instead of using the trick they use in Wikipedia.

[Igor Khavkine]

This may or may not be useful, but when $Y=\mathbb{R}$ (and by simple extension $\mathbb{R}^n$) Lipschitz functions separate points (McShane's extension theorem) and hence by Stone-Weierstrass they are dense in continuous functions. This paper also proves that density holds in uniformly continuous functions provided X and Y allow for McShane-like extensions.

[Yury G. Kudryashov]

If we want Peano existence theorem for Banach spaces, then it's not enough...

[Igor Khavkine]

In that case you may be right. When $X=H_1$ and $Y=H_2$ are Hilbert spaces, there's Kirszbraun's theorem. But quoting from that encyclopedia article:

With the exception of the trivial case when the target is $\mathbb{R}$, generalizations of Kirszbraun's theorem are rather delicate: it is for instance known that it does not hold if any of the two spaces $H_1$ and $H_2$ are replaced by Banach spaces.

[Igor Khavkine]

Yury G. Kudryashov said:

If we want Peano existence theorem for Banach spaces, then it's not enough...

BTW, not sure what scope you were aiming for, but it seems that Peano's existence theorem is in general false for infinite dimensional Banach spaces.

[Yury G. Kudryashov]

Thank you! I was going to check the details some day before starting to formalize the dependencies (the source I was reading proved Peano for finite dimensional spaces, I did hope that the proof can be generalized).

[Antoine Chambert-Loir]

https://mathoverflow.net/a/163713

[Antoine Chambert-Loir]

And, for positive results, see another answer https://mathoverflow.net/a/163737

[Yury G. Kudryashov]

I guess, this means that I can formalize Peano's theorem for finite-dimensional spaces using the trick on Wikipedia and we still need the existing Picard-Lindelöf (existence part), because it works in Banach spaces.

[Yury G. Kudryashov]

Thank you all!