If $f, g$ are two functions on a bounded subset of $\mathbb R$, is there a bound on $\|f-g\|_2$, involving only $\|f-g\|_1$, $\|g\|_2$, and some other finite quantities? Here, $\|\cdot\|_p$ is the $L^p$-norm.
Thanks!
If $f, g$ are two functions on a bounded subset of $\mathbb R$, is there a bound on $\|f-g\|_2$, involving only $\|f-g\|_1$, $\|g\|_2$, and some other finite quantities? Here, $\|\cdot\|_p$ is the $L^p$-norm.
Thanks!
There cannot be. Consider the following functions defined on $[0,1]$.
$f_n(x)=\begin{cases} n&0\leq x\leq 1/n\\ 0& 1/n
$g(x)=0$
Then $\|f_n-g\|_1=1$ for all $n$ but $\|f_n-g\|_2$ explodes.