Lipschitz-free Banach spaces
We show that when a linear quotient map to a separable Banach space $X$ has a Lipschitz right inverse, then it has a linear right inverse. If a separable space $X$ embeds isometrically into a Banach space $Y$, then $Y$ contains an isometric linear copy of $X$. This is false for every nonseparable weakly compactly generated Banach space $X$. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space $X$ has the bounded approximation property and $Y$ is Lipschitz isomorphic to $X$, then $Y$ has the bounded approximation property.