Extreme non-differentiability of typical Lipschitz mappings

<jats:p> We show that no matter what subset of a normed space is given, a typical 1-Lipschitz mapping into a Banach space is non-differentiable at a typical point of the set in a very strong sense: the set of partial limits of its derivative ratios, which is separable, contains all linear operators of norm at most <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mn>1</mml:mn> </mml:math> <jats:tex-math>$1$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S2050509426102369_inline1.png"> <jats:alt-text content-type="machine-generated">1</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> from any fixed separable subspace of the operator space. </jats:p> <jats:p> For subsets of finite-dimensional normed spaces which can be covered by a countable union of closed purely <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mn>1</mml:mn> </mml:math> <jats:tex-math>$1$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S2050509426102369_inline2.png"> <jats:alt-text content-type="machine-generated">1</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> -unrectifiable sets this extreme non-differentiability holds for a typical Lipschitz mapping at every point. </jats:p> <jats:p>Both results are new even for Lipschitz mappings with a finite-dimensional codomain.</jats:p>