Let $C^\infty(R^d)$ be the space of infinite order differentiable function space. As well known that $C^\infty(R^d)$ is not separable under uniform norm $$ \|f\|=\sup_{x\in R^d}|f(x)|. $$ However, it seems that $C^\infty(R^d)$ is separable under Hölder norm $$ \|f\|_\alpha=\sup_{x\in R^d}|f(x)|+\sup_{x,y\in R^d,x\ne y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}. $$ I don't know how to prove it.
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What exactly is the space $C^\infty$? Without further assumptions, the norm could be infinite on some elements of this space. – PhoemueX Dec 25 '20 at 14:39
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I am sorry that I forgot to set the space C^\infty(R^d) with the bounded property. – alphabeta Dec 28 '20 at 01:24