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Consider the space $C^{\infty}[a,b]$ with norm $||f|| = max_{[0,1]} |f(x)|$, with $f\in C^{\infty}[a,b]$. Is the differentiation operator $\frac{d}{dx}$ continuous on $C^{\infty}[a,b]$?

I'm very confused because it seems almost trivial-- there are plenty of examples of the derivative not being a continuous operator-- that's why it's so hard to study.. but in this space, isn't it defined to be only those functions whose derivatives are infinitely differentiable? How do I show this formally?

Martin
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user79015
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2 Answers2

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Probably you don't want just the sup norm on smooth functions. Rather, topologize smooth functions by the family of seminorms given by sups of all derivatives. (This is a Frechet space, the projective limit topology on the Banach spaces $C^k[a,b]$ with sum of sups of derivatives up to order $k$.) Differentiation is continuous in that topology. (And differentiation is continuous from $C^k[a,b]$ to $C^{k-1}[a,b]$.)

A reason to understand that just the sup norm of values is not the right topology on smooth functions is that the space is not complete in that topology.

paul garrett
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Consider $f_n(x) = \sin(n \pi x)$. $||f_n|| = 1$, but $df_n/dx$ is unbounded.

al0
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