Approximate Morse-Sard type results for non-separable Banach spaces (Q6547193)

This interesting article is devoted to Morse-Sard type theorems in non-separable Banach spaces. Roughly speaking, the results assert that, under some conditions on a (non-separable) Banach space \(E\), every continuous map \(f\) from \(E\) to a quotient \(F\) of \(E\) can be uniformly approximated by \(C^k\)-smooth functions without any critical point. Recall that a critical point for a differentiable function \(g\colon E\to F\) is a point \(x\) such that \(g'(x)\colon E\to F\) is not surjective.\N\NThe classical Morse-Sard theorem claims that for a sufficiently differentiable function \(f\colon \mathbb{R}^n\to \mathbb{R}^d\), the image of the set of critical points (namely, the set of critical values) is Lebesgue null in \(\mathbb{R}^d\). However, it is by now well known that such a result does not extend to the infinite-dimensional context. There are even examples (due to \N\textit{S.~M. Bates} and \textit{C.~G. Moreira} [C. R. Acad. Sci., Paris, Sér.~I, Math. 332, No.~1, 13--17 (2001; Zbl 0992.58003)])\Nof homogeneous polynomials of degree \(3\) from \(\ell_2\) to \(\mathbb{R}\) whose set of critical values contains an interval. On the other hand, in the separable, infinite-dimensional context there are several approximation results. For instance, every continuous map from \(\ell_2\) to \(\mathbb{R}^d\) admits uniform approximations by \(C^\infty\)-smooth functions without critical points and the same is true with \(C^1\)-smooth approximations in separable Asplund spaces.\N\NRecently, \textit{D.~Azagra} et al. [Adv. Math. 354, Article ID 106756, 80~p. (2019; Zbl 1434.46050)] managed to obtain similar approximation results for functions \(f\colon E\to F\), where \(F\) is an infinite-dimensional Banach space. Notice that in this case a necessary condition is that \(F\) is a quotient of \(E\). In particular, it is proved there that every continuous function \(f\colon E\to F\), where \(E=c_0, \ell_p, L_p\) with \(p\in(1,\infty)\) and \(F\) is a quotient of \(E\), can be uniformly approximated by \(C^k\)-smooth functions without critical points.\N\NThe present paper contains the first results available outside the separable context, in particular addressing non-separable Hilbert spaces for the first time. In particular, in Section~3 the authors prove that if \(E\) is \(c_0(\Gamma)\) or \(\ell_p(\Gamma)\) with \(p\in(1,\infty)\) and \(F\) is a quotient of \(E\), then every continuous function from \(E\) to \(F\) admits a uniform approximation by \(C^k\)-smooth functions without critical points (where \(k\) depends on the value of \(p\)). Subsequent sections of the paper then prove more general and technical results. The paper is well written, with detailed references to the literature and careful explanation of the main ingredients required in the arguments, first illustrated in the particular (yet highly non-trivial) case of the long sequence spaces \(c_0(\Gamma)\) and \(\ell_p(\Gamma)\).