Riesz Representation Theorem for $\ell^p$

Let $ 1 \leq p < \infty$, with $q$ the conjugate of $p$, and let $T \in \ell^{p*}$. Then for some sequence $g \in \ell^q,$ $T(f)=\sum_{\mathbb{N}} fg$ for all $f \in \ell^p$.

I am trying to prove this from elementary principles, so no referring to counting measures and deriving the result as an appendage of more general versions of the Riesz Representation Theorem. I do, however, have the RRT for $L^p$ at my disposal, but I can't make heads or tails of trying to extend an arbitrary functional on $\ell^p$ to one on $L^p$.

My approach is, then, to force the theorem by defining $g=\sum T(e_n)$ where $e_n$ is the sequence (in $L^p$) with a $1$ in the $n$th place and zero otherwise. Then if $f=\sum a_k e_n$, $$T(f)=\sum a_k T(e_n)$$ by linearity, and this is bounded as $T$ is bounded. Hence for all $f \in \ell^p$, $$\sum f(k)T(e_n) \in \ell^1$$

Here's where I hit the snag. How can I conclude $T(e_n) \in \ell^q$, without begging the question and using the RRT? I've attempted writing some function in $l^p$ from it, something like $T(e_n)^{(q-1)}*\frac{1}{n}$, and proving this part by contradiction, but I can't get anything to work. Thanks in advance for any advice!

Solution 1:

If you already know $(L^p)^*=L^q$, you can consider the isometric embedding $J:\ell^p\to L^p(0,\infty)$ that sends basis vectors $e_k$ to $\chi_{[k-1,k]}$. Every linear functional on $\ell^p$ can be viewed as a functional on $\operatorname{ran}J$, and from there extended (Hahn-Banach) to all of $L^p$. This gives a representation of the functional as some $g\in L^q(0,\infty)$. Finally, let $y_k = \int_{k-1}^k g\,dx$ and note that $(y_k)\in \ell^q$.

However, the above circuitous proof may be not worth the trouble. It is easier to get over the snag that you hit. To this end, consider truncated sequence $$g = (T(e_1),T(e_2),\dots, T(e_N), 0,0,\dots)$$ which is certainly in $\ell^q$. If you can show that $\|g \|_q$ is bounded independently of $N$, the conclusion $(T(e_n))\in \ell^q$ will follow.

Taking $x_n = |T(e_n)|^{q-2}T(e_n)$ you will find that $\|x\|_p=\|g \|_q^{q-1}$ and $Tx=\|g\|_q^q$. Hence, $$ \|g\|_q^q \le \|T\|_{(\ell^p)^*} \|g \|_q^{q-1} $$ and the conclusion $\|g\|_q \le \|T\|_{(\ell^p)^*}$ follows.