The following is motivated by Problem 2 in Paul Halmos' "Hilbert Space Problem Book" (my edition is from 1967 by American Book - Van Nostrand - Reinhold, thanks to Ralf Pradella for lending me the book). The Problem 2 says "Find a coordinatized proof of the Riesz representation theorem" i.e. the fact that each linear continuous functional on a Hilbert space is represented by the scalar product with a constant vector in the same Hilbert space. I would like to show a similar argument on a more general $l^p, 1< p< \infty$ instead of $l^2$. My question is how does this relate to the proofs in textbooks.
Let $l^p = l^p(I)$ be the set of complex-valued sequences $a:I\rightarrow C$ with coordinates $a_i \in \mathbb{C} $ on a discrete measure space $I$ with points $i\in I$ of mass 1 as atoms, and norm $\left\lVert a \right\rVert_p = (\sum _{i \in I} |a_{i}|^p)^{1/p}<\infty$. Countability of the underlying space $I$ is not critical and hence not assumed, so our "sequences" are rather generalized sequences. (Ultimately, the support of a linear functional turns anyway out to be a countable index set, so this generalization is not essential.) Unit vectors $e_i\in l^p$ are characteristic functions on the point masses $i\in I$ with value 1 on ${i}$, zero elsewhere.
Take a linear continuous functional $f:l^p \rightarrow \mathbb{C}$ in the dual space of $l^p$, and write $f_i = f(e_i)$. We consider sequences $a_J=\sum_{i\in J}a_ie_i$ with only $finitely$ many non-zero coordinates on a subset $J\subset I$ and take the abstract definition $\left\lVert f \right\rVert$ of the functional norm on duals of normed spaces as follows $$|f(a)|=|f(\sum _{i \in I}a_ie_i)|=|\sum _{i \in I}a_if(e_i)|\leq \left\lVert a \right\rVert_p \left\lVert f \right\rVert$$ Here $\left\lVert f \right\rVert$ is the infimum of all real numbers for which such an inequality holds for all $a\in l^p(I)$. The sequences with finitely many non-zero coordinates are dense in $l^p(I)$, so by linearity and continuity the values $f(e_i)$ determine the functional $f$ uniquely and the inequality holds for all $a\in l^p(I)$.
From Hölder's inequality applied to conjugate exponents $p, q, 1/p+1/q=1$ it is known that every sequence $\{f_i\}\in l^q(I)$ with finite conjugate norm $\left\lVert \{f_i\} \right\rVert_q = (\sum _{i \in I} |f_{i}|^q)^{1/q} < \infty$ determines a continuous linear functional, and the above inequality holds with the concrete value $\left\lVert \{f_i\} \right\rVert_q$ replacing the abstract functional norm $\left\lVert f \right\rVert$. The essence of Riesz-Fischer for $l_p$ and $l_q$ is to show the converse i.e. for every continuous linear functional $f:l^p(I)\rightarrow \mathbb{C}$ exists a corresponding "concrete" sequence $\{f_i\}\in l^q(I)$ and $\left\lVert \{f_i\} \right\rVert_q = \left\lVert f \right\rVert$.
We mimick Halmos' $l^2$ argument as follows in $l^p$ and $l^q$: for every finite subset $J\subset I$ we build a vector $b_J=\sum _{j \in J}b_je_j$ setting $b_j=|f(e_j)|^{1/(p-1)}e^{-i\phi_j}$. Here $e^{i\phi_j}$ are the phases of $f(e_j)=|f(e_j)|e^{i\phi_j}$ as a complex number. Then $$f(b_J)=\sum _{j \in J}b_jf(e_j)=\sum _{j \in J}|f(e_j)|^{1/(p-1)}e^{-i\phi_j}|f(e_j)|e^{i\phi_j}=\sum _{j \in J}|f(e_j)|^{p/(p-1)}=\left\lVert b_J \right\rVert_p^p\leq \left\lVert f \right\rVert \left\lVert b_J \right\rVert_p$$ and from there $$\left\lVert b_J \right\rVert_p\leq \left\lVert f \right\rVert^{1(p-1)}$$
At this point it was intriguing (at least for me) to try further with $\left\lVert b_J \right\rVert_p$, but Halmos does something different and introduces a new functional $f_J$ by setting $f_J(e_j) = f(e_j)$ for $j\in J$ and $f_J(e_i)=0$ for all other $i \notin J$. With this the last 2 inequalities combined yield $$\left\lVert f_J \right\rVert_q^q= \sum _{j \in J}|f(e_j)|^q=\sum _{j \in J}|f(e_j)|^{p/(p-1)}=f(b_J) \leq \left\lVert f \right\rVert \left\lVert b_J \right\rVert_p \leq \left\lVert f \right\rVert \left\lVert f \right\rVert^{1(p-1)}= \left\lVert f \right\rVert^q$$ and thus $$\left\lVert f_J \right\rVert_q \leq \left\lVert f \right\rVert$$
Arguing now this is a uniform bound for $\sum _{j \in J}|f(e_j)|^q$ across all finite subsets $J\subset I$, and the finite subsets build a directed set with inclusion as direction, the net $\{f_J\}$ converges weakly and strongly, the limit $\lim_J f_J$ is equal to $f$ with $\sup_J \left\lVert f_J \right\rVert_q = \left\lVert f \right\rVert$ (to justify equality in the latter, take again the $b_J$ as test functions).
I have not found this line of reasoning in the textbooks available to me, but my knowledge of the standard literature is quite limited and it might be actually a widely popular approach.
Could somebody highlight a reference to me from the mathematical literature? Many thanks in advance.
Only after having posted my question I have just found the following 2 pages here on math.stackexchange which both explain the "coordinatized" approach very well, only the notation being a bit different from mine. So my question can be considered answered.
Riesz Representation Theorem for $\ell^p$
Proving the Riesz Representation Theorem for $\ell^p$.