It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum\_{n = 0}^\infty a\_n T\_n $$ which is absolutely convergent under the $\lVert \, \cdot \, \rVert\_\infty$ norm (see Approximation Theory and Approximation Practice by Lloyd Trefethen, 2018, p.19-20). I'm wondering whether the series converges under the Lipschitz norm, i.e. $$ \Big\lVert \sum\_{n = N}^\infty a\_n T\_n \Big\rVert\_\text{Lip} \xrightarrow{N \to \infty} 0 $$
(and if so, is convergence unconditional? absolute?) where the Lipschitz norm is defined by $$\lVert f \rVert\_\text{Lip} := |f(-1)| + \sup\_{x \neq y}\bigg|\frac{f(y)-f(x)}{y-x} \bigg|$$
I'm interested in this since $(\text{Lip}[-1,1], \lVert \, \cdot \, \rVert\_\text{Lip})$ is a Banach space, while $(\text{Lip}[-1,1], \lVert \, \cdot \, \rVert\_\infty)$ is not. Thank you.



Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?
No, the space of Lipschitz functions on an infinite metric space is non-separable so it can't have a Schauder basis.


