Let $X=L^2(\mu)\times L^2(\mu)=\{(f,g)|f,g\in L^2(\mu)\}$ be the linear space normed by $\|(f,g)\|=(\|f\|_2^3+\|g\|_2^3)^{1/3}$. Show that $X$ is Banach space and describe $X^*$.
My Work:
We have to show that $X$ is complete. So, must prove that every absolutely convergent series is convergent. Suppose $\displaystyle \sum_{n=1}^\infty \|(f_n,g_n)\|$ converges. Then $\|(f_n,g_n)\|\rightarrow 0$ as $n\rightarrow 0$. That is $(\|f_n\|_2^3+\|g_n\|_2^3)^{1/3}\rightarrow 0$ as $n\rightarrow 0$. This implies $(\|f_n\|_2^3+\|g_n\|_2^3)\rightarrow 0$ as $n\rightarrow 0$. Hence $(f_n,g_n)\rightarrow 0$ as $n\rightarrow 0$. After that I was stuck. I want to show that $\displaystyle \sum_{n=1}^\infty (f_n,g_n)$ converges. can somebody please help me to prove it?
Is there anything I can do here with the fact that $L^2(\mu)$ complete?