3
$\begingroup$

Define $C([a,b], \mathbb R)$ to be the space of continuous functions $f : [a,b] \to \mathbb R$ with the norm $\| \cdot \|_{\infty}$. Let $H : C([a, b], \mathbb{R}) \rightarrow \mathbb{R}$ be the map such that $H(f)$ is the definite integral of $f$ from $a$ to $b$ for any $f \in C([a,b], \mathbb R)$.

How do I prove that $H$ is continuous?

Note [SN]: Edited the question to make it more complete.

  • 2
    I guess the $C([a,b], \mathbb R)$ space is endowed with the sup-norm. If so, it might be$a$good idea to mention that in the question. (Also, I think the usual notation is $C([a,b], \mathbb R)$, not $\mathbb C([a,b], \mathbb R)$.)2011-11-04

1 Answers 1

6

Yes, this is because it is Lipschitz:

$ |H(f)-H(g)|=\left|\int_a^b(f-g)\; dx\right|\leqslant |b-a| \cdot \|f-g\|_{\infty} .$

  • 0
    Very grateful actually, I couldn't figure out how to center things, thanks!2011-11-04