Possible Duplicate:
$\lim_{n\rightarrow \infty}\int_0^1f_nhdm=\int_0^1fhdm$, prove $f\in L^p(m)$ , where $1\le p<\infty$.
Can anyone help with this question?
When ${f_n}$ is defined on [0,1], $ ||f_n||_p\le1$, $1
Prove $f\in L_p$
Possible Duplicate:
$\lim_{n\rightarrow \infty}\int_0^1f_nhdm=\int_0^1fhdm$, prove $f\in L^p(m)$ , where $1\le p<\infty$.
Can anyone help with this question?
When ${f_n}$ is defined on [0,1], $ ||f_n||_p\le1$, $1
Prove $f\in L_p$
Since $L^\infty[0,1]$ is dense in $L^q[0,1]$, $$ \|f\|_p=\sup\{\int_0^1fg\,dm: \|g\|_q=1\}=\sup\{\int_0^1fg\,dm: g\in L^\infty[0,1]\mbox{ and }\|g\|_q=1\}. $$ Note that, for $g\in L^\infty$ with $\|g\|_q=1$, $$ \left|\int_0^1fg\,dm\right|=\lim\,\left|\int_0^1f_ng\,dm\right|\leq \|f_n\|_p\,\|g\|_q\leq1. $$ So $\|f\|_p\leq1$.