Is there an example of a sequence $\{f_n\}$ in $L^1(\mathbb{R})$, such that:
- $\{||f_n||_1\}$ is bounded.
- There's a convergent subsequence $f_{\phi(n)}$, i.e. $\exists f \in L^1(\mathbb{R})$ such that $||f_{\phi(n)} - f||_1 \longrightarrow 0$.
- If there's another subsequence that converges, then it converges to $f$.
But $f_n$ does not converge in $L^1(\mathbb{R})$.
What about in $L^p(\mathbb{R})$?
EDIT: Would this sequence work?
$f_{2n}=0$
$f_{2n+1}=1_{[n, n+1]}$
Put $g_n = \chi[n,n+1) \in L^P(\mathbb R)$ for $n \in \mathbb N,$ i.e. $g_n$ is the characteristic function of the interval $[n,n+1).$ Then we have $\|g_n\|_p = 1.$ Now, define $f_{2n} = g_{2n}$ and $f_{2n-1} = 0$ for $n \in \mathbb N,$ i.e. $f_n$ is alternatingly $0$ or $g_n$. Note that $\|g_a-g_b\|_p \geq c > 0$ for $a\neq b$ with an appropriate $c = c(p)$ depending on $p$. This implies that for any convergent subsequence $h_m$ of $f_n$, we must have $h_m = 0$ for all $m$ large enough, since the $h_m$ must be Cauchy in $L^p(\mathbb R).$ Yet, $f_n$ doesn't converge, since, for example, $\|f_n\|_p$ doesn't converge.