Let $(\Omega,d)$ be a non complete metric space, and consider the space $C^{\gamma}_b(\Omega, \mathbb{R})$, $0<\gamma<1$ of all bounded Hölder continuous functions.

Is $C^{\gamma}_b(\Omega, \mathbb{R})$ endowed with the norm $$ \|f\|_{\gamma}=\|f\|_0+\sup_{x\neq y} \dfrac{|f(x)-f(y)|}{d(x,y)^{\gamma}} $$

a Banach Space?

Could someone provide me some reference?


1 Answer 1


Yes. Let $$\|f\|_{C^\gamma_b} = \sup\limits_{x \in \Omega}|f(x)| + \sup\limits_{x,y \in \Omega, x \neq y} {|f(x) - f(y)| \over d(x,y)^\gamma}.$$ You can easily check that this is a norm. To see that it is complete, suppose that $f_n \in C^\gamma_b$ is a Cauchy sequence. Then $f_n$ is Cauchy with respect to the $\sup$ norm, so there exists $f \in C_b$ such that $f_n$ converges uniformly to $f$. We want to show that in fact $f \in C_b^\gamma$, and that $f_n \to f$ in $C_b^\gamma$ norm. Let $\epsilon > 0$, and choose $N>0$ such that, if $n,m \geq N$, $\|f_n - f_m\|_{C_b^\gamma} \leq \epsilon/3$. Let $n \geq N$, and let $x,y \in \Omega$, $x \neq y$. Choose $L > n$ such that $|f_L(x) - f(x)| \leq {\epsilon \over 3 d(x,y)^\gamma}$ and $|f_L(y) - f(y)| \leq {\epsilon \over 3 d(x,y)^\gamma}$. We then have $${|(f_n(x) - f(x)) - (f_n(y) - f(y))| \over d(x,y)^\gamma} \leq {|(f_n(x) - f_L(x)) - (f_n(y) - f_L(y))| \over d(x,y)^\gamma}$$ $$+ {|f_L(x) - f(x)| \over d(x,y)^\gamma} + {|f_L(y) - f(y)| \over d(x,y)^\gamma} \leq \epsilon.$$ Since this holds for every $x$ and $y$, we have $$\sup\limits_{x,y \in \Omega, x \neq y} {|(f_n - f)(x) - (f_n - f)(y)| \over d(x,y)^\gamma} \leq \epsilon,$$ and therefore $f_n \to f$ in $C_b^\gamma$. The above also clearly implies that $f \in C_b^\gamma$, since $$\sup\limits_{x,y \in \Omega, x \neq y} {|f(x) - f(y)| \over d(x,y)^\gamma} \leq \|f - f_n\|_{C_b^\gamma} + \sup\limits_{x,y \in \Omega, x \neq y} {|f_n(x) - f_n(y)| \over d(x,y)^\gamma} < \infty.$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.