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Let $c_0$ be the subspace of $\ell^\infty$ consisting of sequences that converge to $0$. Show that $c_0$ is a closed subspace of $\ell^{\infty}$ whose dual space is isomorphic to $\ell^1$. Conclude that $c_0$ is not reflexive and therefore neither is $\ell^{\infty}$.

The final part seems clear to me, the one regarding reflexivity. For the first part, about closedness, I am not sure which of the methods is appropriate. Would showing that the complement is open work or is there something better? For the isomorphism part I have no idea.

Thank you very much in advance for any help.

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1 Answer 1

As for the closedness. Let $x\in\ell_\infty$ be the limit point of $\{x_n:n\in\mathbb{N}\}\subset c_0$, i.e. $x=\lim\limits_{n\to\infty} x_n$ in $\ell_\infty$. Since convergence in $\ell_\infty$ is the uniform convergence on $\mathbb{N}$ we can "interchange limit signs". More preciesly $$ \begin{align} \lim\limits_{k\to\infty}x(k)&=\lim\limits_{k\to\infty}\lim\limits_{n\to\infty} x_n(k)\\ &=\lim\limits_{n\to\infty}\lim\limits_{k\to\infty} x_n(k)\\ &=\lim\limits_{n\to\infty}0\quad (\text{ since } \{x_n:n\in\mathbb{N}\}\subset c_0)\\ &=0 \end{align} $$ Hence $x\in c_0$. Since $x$ was choosen too be arbitrary limit point of $c_0$, we conclude that $c_0$ is closed.

As for duality. Consider map $$ I:\ell_1\to (c_0)^*:x\mapsto\left(y\mapsto\sum\limits_{k=1}^\infty x(k)y(k)\right) $$ We need to prove that $I$ is a surjective isometry. Take $x\in\ell_1$, then for all $y\in c_0$ we have $$ \begin{align} |I(x)(y)|&=\left|\sum\limits_{k=1}^\infty x(k)y(k)\right| \leq\sum\limits_{k=1}^\infty |x(k)y(k)| \leq\sum\limits_{k=1}^\infty \sup\{|x(k)|:k\in\mathbb{N}\}|y(k)|\\ &=\sum\limits_{k=1}^\infty \Vert x\Vert |y(k)| =\Vert x\Vert\sum\limits_{k=1}^\infty|y(k)|=\Vert x\Vert\Vert y\Vert \end{align} $$ Hence $\Vert I(x)\Vert\leq \Vert x\Vert$. Now fix $\varepsilon>0$, then there exist $K\in\mathbb{N}$ such that $\sum\limits_{k=1}^K|x(k)|>\Vert x\Vert-\varepsilon$. Then consider $y'\in c_0$ defined by $$ y'(k)= \begin{cases} \overline{x(k)}|x(k)|^{-1}&\quad\text{ if }\quad k\leq K\\ 0&\quad\text{ otherwise} \end{cases} $$ In this case $\Vert y'\Vert=1$ and $$ \begin{align} |I(x)(y')|&=\left|\sum\limits_{k=1}^\infty x(k)y'(k)\right| =\sum\limits_{k=1}^K x(k)\overline{x(k)}|x(k)|^{-1} =\sum\limits_{k=1}^K |x(k)| =\sum\limits_{k=1}^K |x(k)|\Vert y'\Vert\\ &\geq (\Vert x\Vert-\varepsilon)\Vert y'\Vert\\ \end{align} $$ Hence $\Vert I(x)\Vert\geq\Vert x\Vert-\varepsilon$. Since $\varepsilon$ is arbitrary we conclude $\Vert I(x)\Vert\geq\Vert x\Vert$. On the other hand as we proved earlier $\Vert I(x)\Vert\leq \Vert x\Vert$, hence $\Vert I(x)\Vert=\Vert x\Vert$. Since $x$ is arbitrary, then $I$ is an isometry.

Let's show that $I$ is surjective. For each $n\in\mathbb{N}$ we denote by $e_n$ the vectrofrom $c_0$ defined by $$ e_n(k)= \begin{cases} 1&\quad\text{ if }\quad k=n\\ 0&\quad\text{ otherwise } \end{cases} $$ Now take arbitrary $f\in(c_0)^*$ and consider $x'\in\ell_\infty$ defined by $x'(k)=f(e_k)$. Fix $K\in\mathbb{N}$ and consider vector $y''\in c_0$ defined by $$ y''(k)= \begin{cases} \overline{f(e_k)}|f(e_k)|^{-1}&\quad\text{ if }\quad k\leq K\\ 0&\quad\text{ otherwise } \end{cases} $$ Obviously $\Vert y''\Vert=1$. Moreover $$ \sum\limits_{k=1}^K |x'(k)|=\sum\limits_{k=1}^K f(e_k)y''(k)=f\left(\sum\limits_{k=1}^K y''(k)e_k\right)=f(y'')=|f(y'')|\leq\Vert f\Vert\Vert y''\Vert=\Vert f\Vert $$ Since $K\in\mathbb{N}$ is arbitrary $$ \sum\limits_{k=1}^\infty |x'(k)|\leq\Vert f\Vert $$ Hence $x'\in\ell_1$. Fix $\varepsilon>0$ and $y\in c_0$. Then there exists $K\in\mathbb{N}$ such that $k>K$ implies $|y(k)|<\varepsilon$. This is equivalent to say that $k>K$ implies $\left\Vert y-\sum\limits_{n=1}^k y(n)e_n\right\Vert<\varepsilon$. Since $\varepsilon$ is arbitrary this is equivlent to $y=\lim\limits_{n=1}^\infty y(n)e_n$. Then we may say $$ f(y)=f\left(\sum\limits_{n=1}^\infty y(n)e_n\right)=\sum\limits_{n=1}^\infty y(n)f(e_n)=\sum\limits_{n=1}^\infty y(n)x'(n)=I(x')(y) $$ Since $y\in c_0$ is arbitrary, then $f=I(x')$. Since $f\in (c_0)^*$ is arbitrary $I$ is surjective.

Since $I$ isometric and surjective this is isometric isomorphism, hence $(c_0)^*\cong\ell_1$.

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