Equality of expressions containing supremum of some double sums I am doing a project related to operator norm and some double sequences. In the course of proving some results, I encounter the following expressions:
$\displaystyle \|\alpha\| := \sup_{\|x\|_p=1} \left[\sum_{j=1}^\infty \left|\sum_{i=1}^\infty x_i \alpha_{ij}\right|^q\right]^{\frac{1}{q}}$
and
$\displaystyle \|\alpha\|^* := \left(\sum_{j=1}^\infty \sum_{i=1}^\infty |\alpha_{ij}|^q \right)^{\frac{1}{q}}$,
where $\|\cdot\|_p$ is the $\ell_p$ norm, $p$ is the conjugate of $q$, and $x=(x_i)$ is a sequence, $\alpha=(\alpha_{ij})$ is a double-indexed sequence.
I want to say that they are equal. But is it true that $\|\alpha\|= \|\alpha\|^*$ ?
Thank you.

You can take $\alpha_{11}=\alpha_{22}=1$, and all other coefficients equal to $0$. Let us fix $p=q=2$. Then
$$
\|\alpha\|^*=2^{1/2},
$$
while, for $x$ with $\|x\|_2=1$,
$$
\left[\sum_{j=1}^\infty \left|\sum_{i=1}^\infty x_i \alpha_{ij}\right|^2\right]^{\frac{1}{2}}=(|x_1|^2+|x_2|^2)^{1/2}\leq\|x\|_2=1.
$$
So $\|\alpha\|=1$.
