Question on definition of $\;\alpha-$Holder norms

Question

I recently started to study about Holder norms and here is a definition from Evans' book:

When our professor wrote down these definitions, he said that "We need the $\;L_{\infty}-$norm in order to define Holder norm because constant functions would be a problem".

As I try to understand everything he says, I explained to myself the above statement as follows:

If $\;u(x)=const.\;$ then $\;[u]_{C^{0,γ}}=0\;$ but $\;u(x)\;$ is not necessary zero. Thus it couldn't be $\;[u]_{C^{0,γ}}\;$ the Holder-norm. (And this is why Evans states it as a seminorm)

Is the above thought correct? I would really appreciate if somebody could confirm it and moreover suggest me some books where I could see Holder-norms in more details.

Thanks in advance!

Answer 1

Here are two references:

• Krantz' paper "Lipschitz Spaces, Smoothness of Functions, And Approximation Theory",
• Fiorenza's book Hölder and locally Hölder Continuous Functions, and Open Sets of Class $C^k$, $C^{k,\lambda}$.