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I have been trying to show that $C(X)$ is not strictly convex but I have been having a tough time, any help would be appreciated.

An Invisible Carrot
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  • This depends on $X$. Can you find $f\in C(X)$ that has norm 1 without being constant? – Hagen von Eitzen May 27 '20 at 13:12
  • It's a Rudin Exercise (Chapter 5 Question 3) and we first show that $L^p(\mu)$ is strictly convex for $1<p<\infty$, but it then asks to show that this fails for $p=1,\infty$ and in every $C(X)$. Doesn't define a norm for $C(X)$? – An Invisible Carrot May 27 '20 at 13:13

2 Answers2

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I assume that your $X$ is compact (so that $\|f\|_\infty$ is defined for all continous $f\colon X\to\Bbb R$). If we additionally assume that $X$ allow the existence of non-constant continuous functions (e.g., $X$ is not endowed with the indiscrete topology), then the following works:

Let $f\colon X\to\Bbb R$ be non-constant continouus. Then $\|f\|\ne 0$ and $g:=\frac1{\|f\|}f$ has norm $1$. Then there exists $x_0\in X$ with $g(x_0)=\pm 1$. Define $h(x)=g(x_0)$. Then all convex combinations of $g$ and $h$ have norm $1$.

Hagen von Eitzen
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For $X = \mathbb{R}$, take a function $f$ which is $0$ before $-2$, strictly increase until $-1$ where $f(-1)=1$, which is constant until $0$, and is even. Take $g(x) = f(x-1)$. Then $f \neq g$, $\sup |f| = \sup|g| = \sup \dfrac{|f+g|}{2}=1$, thus the unit ball in $C(\mathbb{R})$ is not strictly convex.

If you can construct such similar functions on $X$, then the answer will be the same

Didier
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  • Thanks, but would this not mean we have infinite such $X$ to look at? Surely this argument must be a function of the space itself? – An Invisible Carrot May 27 '20 at 13:16
  • In fact, if there exists a continuous function on $X$ with compact support and which is equal to $1$ on an open subset, I think (maybe under suitable assumptions again?) this construction will work. But yes, if $X$ is reduced to a point, the result is conversly that the ball is stictly convex. I don't know where to draw the line! – Didier May 27 '20 at 13:19