Proving the Riesz Representation Theorem for $\ell^p$.

Question

State and prove a Riesz Representation Theorem for the bounded linear functionals on $\ell^p$, for $1 \le p < \infty$.

Let $T : \ell^p \to \Bbb{R}$ be a bounded linear functional, and let $q$ be the conjugate of $p$. Let $e_n$ denote the vector with a $1$ in the $n$-th coordinate and $0$'s elsewhere; and let $b_n = T(e_n)$, which makes since $e_n \in \ell^p$. Then given $(a_n)_{n=1}^\infty \in \ell^p$,

\begin{align} T(a_n)_{n=1}^\infty &= T(\sum_{n=1}^\infty a_n e_n) \\ &= \sum_{n=1}^\infty a_n T(e_n) \\ &= \sum_{n=1}^\infty a_n b_n, \\ \end{align}

and so it remains to show that $(b_n)_{n=1}^\infty \in \ell^q$. To do this, it was suggested, in the chat room, that I look for a sequence $(c_n)_{n=1}^\infty$ for which $T(\sum_{n=1}^\infty c_ne_n) = \sum_{n=1}^\infty |b_n|^q$ holds. Well, $c_n = |b_n|^{q-1} sgn(b_n)$ appears to work:

\begin{align} T(\sum_{n=1}^\infty c_ne_n) &= \sum_{n=1}^\infty |b_n|^{q-1} sgn(b_n) T(e_n)\\ &= \sum_{n=1}^\infty |b_n|^{q-1} sgn(b_n)b_n \\ &= \sum_{n=1}^\infty |b_n|^q \\ \end{align}

The only trouble is, in order for $T(\sum_{n=1}^\infty c_ne_n)$ to make sense, it must be the case that $\sum_{n=1}^\infty c_ne_n \in \ell^p$ or that $\infty > \sum_{n=1}^\infty |c_n|^p = \sum_{n=1}^\infty |b_n|^q$...It was then suggested that look applying to partial sequence $(b_1,...,b_n,0,...)$, all of which are obviously in $\ell^p$. I thought that if I could show that $\sum_{n=1}^N c_n e_n \to \sum_{n=1}^\infty c_n e_n$ in $\ell^p$ as $N \to \infty$, then I would be finished. Sadly, this amounts to showing that $\sum_{n=N}^\infty |b_n|^q \to 0$ as $N \to \infty$; and unfortunately, this does not converge to $0$, unless converges the series itself converges, which I initially and mistakenly thought was true.

I could use a hint on how to show that $\sum_{n=1}^\infty c_n e_n \in \ell^p$.

Answer 1

As you said all it remains is to show is $\sum_n b_n e_n \in l^q$ i.e $\sum_n c_n e_n \in l^p$.

As you also said $\sum_{n=1}^N c_n e_n \in l^p$ and $\sum_{n=1}^N |c_n|^p =\sum_{n=1}^N |b_n|^{(q-1)p}= \sum_{n=1}^N |b_n|^{q}$

More over: $$T\left(\sum_{n=1}^N c_n e_n \right)=\sum_{n=1}^N c_n T\left(e_n \right)=\sum_{n=1}^N c_n b_n=\sum_{n=1}^N |b_n|^{q-1+1}$$

But (and this where you use the continuity of $T$) there exist $C$ ($C=\|T\|_{(l^p)^*}$) such that for all $u \in l^p$: $$|T(u)| \leq C \|u\|_{l^p}$$ so: $$\sum_{n=1}^N |b_n|^{q}=\left|T\left(\sum_{n=1}^N c_n e_n \right) \right| \leq C \left( \sum_{n=1}^N |c_n|^p \right)^\frac{1}{p}=C \left(\sum_{n=1}^N |b_n|^{q} \right)^\frac{1}{p}$$ i.e: $$\left(\sum_{n=1}^N |b_n|^{q} \right)^{\left(1-\frac{1}{p}\right)}=\left(\sum_{n=1}^N |b_n|^{q} \right)^\frac{1}{q} \leq C $$ taking $N \to + \infty$ you obtain: $$\left(\sum_{n=1}^\infty |b_n|^{q} \right)^\frac{1}{q} \leq C$$

Answer 2

Is the following approach feasible? Picking up from

\begin{align} T(\sum_{n=1}^\infty c_ne_n) &= \sum_{n=1}^\infty |b_n|^{q-1} sgn(b_n) T(e_n)\\ &= \sum_{n=1}^\infty |b_n|^{q-1} sgn(b_n)b_n \\ &= \sum_{n=1}^\infty |b_n|^q \\ \end{align}

we want to show $|| \{b_n\}^{\infty}_{n=1} ||_q \in \ell^q$. By boundedness of T,

$$ \left| \;T( \sum_{n=1}^{\infty} c_n e_n) \; \right | \leq ||T||_{*} \cdot || \{c_n\}^{\infty}_{n=1} ||_p $$

where $||T||_{*}$ is the essential upper bound of T.

By Holder's inequality,

$$ \left|\left| \;T( \sum_{n=1}^{\infty} c_n e_n) \; \right| \right|_1 \leq || \{c_n\}^{\infty}_{n=1} ||_p \cdot || \{b_n\}^{\infty}_{n=1} ||_q$$

Therefore $||T||_{*} = || \{b_n\}^{\infty}_{n=1} ||_q $ and since $||T||_{*} < \infty$, $|| \{b_n\}^{\infty}_{n=1} ||_q$ is also bounded so $\{b_n\}^{\infty}_{n=1} \in \ell^q $.