Example for $\|f_n\|_p = 1$ and $\|f_n - f_m\|_p = 1$ for $m \neq n$ Let $X=[0,1]$ with Lebesgue measure. Let $p = [1,\infty]$. Give an example of $f_n$ such that $\|f_n\|_p = 1$ for all $n$ and $\|f_n - f_m\|_p = 1$ for $m \neq n$.

I find it hard to guarantee $\|f_n - f_m \| = 1$. Anyone has an idea how to make it happen?
Construct such a sequence in $L^p(1,\infty)$, say $g_n(t)=2^{-1/p}1_{(n,n+2)}(t)$ if $n=1,2$ and $g_n(t)=2^{-1/p}( 1_{(2,3)}(t) +1_{(n+1,n+2)}(t))$, and then put $f_n(t)=t^{-2/p}g_n(1/t)$.

To see that the norm of the difference is one just calculate
$$
\int_0^1 |f_n-f_m|^p = \int_0^1 t^{-2}|g_n(1/t)-g_m(1/t)|^p dt =\int_1^\infty |g_n(s)-g_m(s)|^p ds =\| g_n-g_m\|_p^p ,
$$
where we used the change of variables $s=1/t$.
Given $p=\infty$, consider

$$f_n = 1_{\{\frac{1}{2^{n}}&lt;x\leq \frac{1}{2^{n-1}}\}}$$