Does the Skorokhod Banach space $D[0,1]$ (cadlag functions equipped with the uniform norm) admit a smooth partition of unity? I found [Johanis - Smooth partitions of unity on Banach spaces](https://www.sciencedirect.com/science/article/abs/pii/S0022123617301398), which provides several classes of Banach spaces with this property, but don't see how to relate the given criteria to the Skorokhod space.



Does the Skorokhod space with the uniform topology admit a smooth partition of unity?
The Skorohod space $D$ has $C([0,1])$ as a Banach subspace that does not have smooth bump functions by results of Bonic and Frampton from 1965. If $D$ had smooth partitions of unity, it would posses smooth bump functions, hence also $C([0,1])$ which is false. So the answer is **no**.


See Section 14 (pp. 152−158, in particular 14.11(1)) in Kriegl and Michor's *The Convenient Setting of Global Analysis* (AMS 1997) for an account on bump functions.


