Assume that $(a_{ij})_{i,j=1}^{\infty}$ is a matrix satisfying the following condition $$\sum_{i,j=1}^{\infty} |a_{ij}|^q<\infty \ (q>1),$$ where $x=\{\xi_j \}_{j=1}^{\infty} \in l_p$, $\frac{1}{p}+\frac{1}{q}=1$, we define $$Ax=\{ \eta_i \}_{i=1}^{\infty},$$ where $$\eta_i=\sum_{j=1}^{\infty} a_{ij}\xi_j, \ (i=1,2,\ldots),$$ Compute $\|A \|$.
Using Holder inequality we can obtain $$\| Ax\| \leq \|x\| \left(\sum_{i,j=1}^{\infty} |a_{ij}|^q \right)^{1/q}$$ Now, I want to choose $x_0 \in l_p$ such that $\|x_0 \|=1$ and $$\|Ax_0 \|\geq \left(\sum_{i,j=1}^{\infty} |a_{ij}|^q \right)^{1/q}$$ Could you please give me some suggestions for $x_0$? Thank you so much.