The space $C^0([0;1],\mathbb{R})$ of all continuous, real-valued functions on $[0,1]$ is not reflexive.

How does one prove that $C^0([0;1],\mathbb{R})$ equipped with the supremum norm is not reflexive?

I don't understand how to show that the $J$ mapping is not surjective.

Solution 1:

As we have seen in a previous question, if $E$ is a reflexive Banach space then each linear continuous functional attains its norm. So, in order to show that $E:=C^0([0,1],\Bbb R)$ endowed with the supremum norm is not reflexive, it is enough to find a linear functional which is not norm attaining. We can define $$x'(f):=\int_0^{1/2}f(t)\mathrm dt-\int_{1/2}^1f(t)\mathrm dt;$$ $x'$ is linear and $|x'(f)|\leqslant \lVert f\rVert_{\infty}$, hence $x'$ is continuous and its norm is $\leqslant 1$.

To see that the norm is indeed $1$, for $n$ integer, consider a function $f_n$ which is $1$ on $[0,1/2-1/n)$ and $-1$ on $(1/2+1/n,1)$, and linear on $(1/2-1/n,1/2+1/n)$. We can see that $\lVert f_n\rVert=1$ and $x'(f_n)=1-2/n$.

Now, we have to show that we cannot find $f\in E$ such that $x'(f)=1$ and $\lVert f\rVert=1$. Let $f$ be continuous of norm $1$. We have to show that $x'(f)\neq 1$, for a fixed $\varepsilon>0$ we can find $\delta>0$ such that $|f(t)-f(1/2)|\leqslant \varepsilon$ whenever $|t-1/2|\leqslant \delta$. Since $$\tag{*} x'(f)=\int_0^{1/2-\delta}f(t)\mathrm dt+\int_{1/2-\delta}^{1/2+\delta}f(t)\mathrm dt -\int_{1/2+\delta}^1f(t)\mathrm dt,$$ we can assume that $f(x)=1$ on $[0,1/2-\delta]$ and $f(x)=-1$ on $[1/2+\delta,1]$, otherwise it is clear that $x'(f)\neq 1$.

By (*), it follows that $|x'(f)|\leqslant 1-2\delta+2\delta\varepsilon+\delta| f(1/2)|$. Taking $\varepsilon\lt 1-|f(1/2)|$, we get that $|x'(f)|\lt 1$.

Solution 2:

Closed subspaces of reflexive spaces are reflexive but you can find a closed subspace of $C[0,1]$ isomorphic to $c_0$: Choose a sequence of continuous peak functions $f_n:[0,1]\to [0,1]$ with $f(1/n)=1$ and disjoint supports. Then $c_0 \to C[0,1]$, $(x_n)_{n\in\mathbb N} \mapsto \sum\limits_{n=1}^\infty x_nf_n$ is an isometry.