Given a nonzreo $x \in \ell^p$, where $1<p<\infty$, find the unique linear functional $f:\ell^p \to \mathbb C$, such that $|f(x)| = \|x\|_p$ and $\|f\| = 1$.
My attempt is nothing but to generalise from the finite dimensional case, i.e $(\mathbb R^n, \|\cdot\|_p)$ and its dual $(\mathbb R^{n*}, \|\cdot\|_q)$, where $\frac{1}{p}+\frac{1}{q} = 1$. I also used the representation of the dual of $\ell^p$ as $\ell^q$, i.e. $f = (f_1,f_2,\dots)$ with $\|f\| = \left(\sum_j|f_j|^q \right)^{1/q}$. So for the finite dimensional case consider the unit sphere $S_p:=\{x\in \mathbb R^n\ |\ \|x\|_p = 1\}$ on which we have the function $f(x) = \sum_j f_j x_j$ that is a restriction of a linear on $\mathbb R^n$. If we require this function to have an extremum at a specific $x \in S_p$ then $df_x = 0$, which is $\sum_j f_j dx_j = 0$, but the $dx_j$ are not independent and so one uses Lagrange multipliers to introduce the constraint $S_p$. Indeed $$ \left( \sum_j |x_j|^p \right)^{1/p} = 1 \implies \sum_j |x_j|^{p-2} x_j d x_j = 0 \implies \sum_j (f_j - \lambda |x_j|^{p-2} x_j) d x_j = 0 $$ and therefore we have $$ f_j = \lambda |x_j|^{p-2} x_j $$ By the requirements that $\|f\|_q = 1$ and $f(x) = \|x\|_p = 1$ we have $\lambda = \|f\|_q/\|x\|_p^{p-1} = 1$, and so I arrive at the end to $$ f = \frac{\|f\|_q}{\|x\|_p^{p-1}} \left( |x_j|^{p-2}\ x_j \right)_{j} \in \mathbb R^{n*}, $$ which I want to generalise by the density of $c_{00}$ in $\ell^p$ and $\ell^q$ to $$ f = \left( |x_j|^{p-2}\ \overline{x_j} \right)_{j \in \mathbb N} \in \ell^q $$ where the bar denotes complex conjugate. It remains to show uniqueness.
My questions are:
- whether this is a valid solution, and
- what is the standard solution using functional analysis to solve it ?. I suppose that it should involve the Hahn-Banach for the existence of $f$, but
- how can one construct the functional explicitly ?
As far as I'm concerned, you've basically constructed the functional explicitly. The only way to express it more explicitly would be to write it as a functional, i.e. a linear map on $\ell^p$, rather than a point in $\ell^q$: $$f(z) = \sum_j z_j \overline{x}_j|x_j|^{p-2}.$$ We have $$f(x) = \sum_j x_j \overline{x}_j|x_j|^{p-2} = \sum_j |x_j|^2|x_j|^{p-2} = \|x\|_p^p = 1,$$ under our assumption that $\|x\|_p = 1$. Given any $z \in \ell^p$ such that $\|z\|_p \le 1$, we have, by Holder, \begin{align*} |f(z)| &= \left|\sum_j z_j \overline{x}_j|x_j|^{p-2}\right| \\ &\le \sum_j |z_j| |x_j|^{p-1} = \sum_j |z_j| |x_j|^{p/q} \\ &\le \left(\sum_j |z_j|^p\right)^{1/p} \left(\sum_j |x_j|^p\right)^{1/q} \\ &= \|z\|_p \cdot 1 \le 1. \end{align*} Together, this is everything you need (uniqueness aside): an explicit form for the functional, a proof that it is bounded with norm less than or equal to $1$, and a proof that the norm is exactly equal to $1$, being achieved at $x$.
Is your working so far valid? The issue is, Lagrange multipliers are good at narrowing down the possible solution set from the whole space to a finite set, but they're not so good at logically establishing that any of the resulting possible solutions are global maxima. With a bit of extra work, I'm sure you could justify why these solutions must be valid solutions, and further expand on why the density of $c_{00}$ does indeed guarantee that the functionals can be extended to the infinite-dimensional case, but it seems not to be worth the effort.
Your work is certainly good scratch work: it lead you intuitively to a solution, and one that can be verified independently without much trouble. This is how I'd go about it too: use the finite-dimensional cases to establish an educated guess as to this linear functional, then prove it explicitly. I wouldn't try to use any heavier machinery here; note that the Hahn-Banach theorem will never be able to produce an explicit formulation for the functionals it creates, it just guarantees that they exist.
To prove uniqueness, as I said in the comments, we can use Minkowski's inequality, which states (for our purposes):
Now, suppose we had functionals $f, g \in \ell^q \approx (\ell^p)^*$ such that $g(x) = f(x) = 1 = \|x\|_p$ and $\|f\|_q = \|g\|_q = 1$ (note that the $q$-norm is the operator norm as a functional on $\ell^p$). Then, $$\left(\frac{f + g}{2}\right)(x) = \frac{f(x) + g(x)}{2} = \frac{1 + 1}{2} = 1,$$ so $\left\|\frac{f + g}{2}\right\|_q \ge 1$. On the other hand, Minkowski's inequality says, $$1 \le \left\|\frac{f}{2} + \frac{g}{2}\right\|_q \le \left\|\frac{f}{2}\right\|_q + \left\|\frac{g}{2}\right\|_q = \frac{1}{2}\|f\|_q + \frac{1}{2}\|g\|_q = \frac{1}{2} + \frac{1}{2} = 1,$$ thus $$\left\|\frac{f}{2} + \frac{g}{2}\right\|_q = \left\|\frac{f}{2}\right\|_q + \left\|\frac{g}{2}\right\|_q,$$ which is equality in Minkowski's inequality, proving that $\frac{f}{2} = k\frac{g}{2}$ or $\frac{g}{2} = k\frac{f}{2}$ for some (real) $k \ge 0$. Taking the norm of both sides, regardless of which equation is true, we see that $$\frac{1}{2} = k\frac{1}{2} \implies k = 1.$$ That is, $f = g$, establishing uniqueness.