For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then,


$
\left\|u\right\|^p\_{W^\mu\_p(U)} = \left\|u\right\|^p\_{W^{\lfloor\mu\rfloor}\_p(U)} + \sum \int\_U \int\_U \frac{|D^\alpha u(x)-D^\alpha u(y)|^p}{|x-y|^{n+p[\mu]}}dxdy
$


where $[\mu]=\mu-\lfloor\mu\rfloor$ and the sum is taken over all multi-indices $\alpha$ with $|\alpha|=\lfloor\mu\rfloor$


This is from Chapter 14 of The mathematical theory of finite element methods
 By Susanne C. Brenner, L. Ridgway Scott.


Does the above hold for $p=\infty$? For example, for $p=\infty$ do we have (or something similar),


$
\left\|u\right\|\_{W^\mu\_p(U)} = \left\|u\right\|\_{W^{\lfloor\mu\rfloor}\_p(U)} + \sup \sup\_U \sup\_U \frac{|D^\alpha u(x)-D^\alpha u(y)|}{|x-y|^{[\mu]}}
$


where the $\sup$ is taken over all multi-indices $\alpha$ with $|\alpha|=\lfloor\mu\rfloor$. Can this be shown by considering the limit of the case $p<\infty$ as $p\rightarrow\infty$?


Thanks in advance. 



Sobolev-Slobodeckij spaces for p=infinity
Yes, it is Hölder spaces and can be regarded as SS spaces for $p=\infty$. Actually, for natural $\mu$ more correct for functions analysis are Zygmund spaces (with differences of th second order in the definition). They are special cases of Besov spaces, which are defined for $0 < p\le \infty$, $\mu\in \mathbb R$. They coincide with corresponding Sobolev-Slobodetskij in some cases. But there are some differences between cases $p<\infty$ and $p=\infty$. For example, Holder spaces are not separable. 


On the second question, shown what? It is a definition for $p=\infty$.


