Let $(X, d)$ be a compact metric and $0 < \alpha < 1$. The space $\mathrm{Lip}^\alpha(X)$ of Hölder functions of order 𝛼 is the Banach space of all functions 𝑓 from 𝑋 into $\mathbb{K}$ such that $\| f\|=\max \{\| f\|_\infty,L(f)\} <\infty$, where
$$L(f)=\sup\{|f(x)-f(y)|/d^\alpha(x,y):x,y\in X, x\neq y\}$$
is the Hölder seminorm of 𝑓. The closed subspace of functions 𝑓 such that
$$\lim\limits_{d(x,y)\to 0}|f(x)-f(y)|/d^\alpha(x,y)=0$$
is denoted by $\mathrm{lip}^\alpha(X)$. We determine the form of all bijective linear maps from $\mathrm{lip}^\alpha(X)$ onto $\mathrm{lip}^\alpha(Y)$ that preserve the Hölder seminorm.
