Weak convergence in $\ell_p$

Question

I have failed to prove the following statement:

Let $1<p< \infty$. Then $f_n \to f$ weakly in $\ell_p$ if and only if $\mathrm{sup} \| f_n \|_p < \infty$ and $f_n \to f$ pointwise.

Any help will be appreciated.

Answer 1

You cannot prove it, because it is wrong. Take $f_n=(0, \dots, 0, 1 , 0, \dots)$, ie. has a one in the $n$th place and zeros otherwise. This converges pointwise to zero (meaning every entry converges to zero) and $\Vert f_n \Vert_p =1$, but $f_n$ does not converge in $\mathcal{l}^p$ to the zero function (as $\Vert f_n - 0\Vert_p = \Vert f_n \Vert_p = 1$).

Answer 2

Suppose $f_n \rightharpoonup f$: use PUB to prove boundedness, and to prove the pointwise convergence, test against the standard basis vectors $e_j = (0,0, \ldots 0, 1, 0,0, \ldots)$ (the $1$ in the $j$-th slot).

Suppose boundedness and pointwise convergence. Fix any $\xi \in \ell^q$, where $q$ is the Holder conjugate of $p$. The action of $\xi$ on $\ell^p$ is (as you know) $$\langle \xi, g\rangle = \sum_i \xi^i g^i.$$

Let $g^m$ denote the truncation of $g$ at height $m$ (after the $m$-th coordinate, all zeroes). Then $$\langle \xi, f_n - f\rangle = \langle \xi, f_n^m - f^m \rangle + \sum_{i>m}\xi^i (f_n^i - f^i)$$.

We can bound the last term using Holder's inequality by $$(\sup_n \|f_n\|_{\ell^p} + \|f\|_{\ell^p})\|\xi - \xi^m\|_{\ell^q}.$$

So, we can pick $m$ large so that $\|\xi - \xi^m\|_{\ell^q}< \varepsilon$, and then with this $m$ fixed, pass to a pointwise limit in $\langle \xi, f_n^m - f^m \rangle$, since this is a finite sum. Hope this helps.