A note on Clarkson's inequalities (Q2770329)

It is proved that if for each \(n\), one has \(1\leq p_n\leq 2\) and the \((p_n,p_n')\) Clarkson inequality holds in each Banach space \(X_n\), then the \((t,t')\) Clarkson inequality holds in \(({\Sigma}_{n=1}^{\infty}X_n)_r\), the \(l_r\)-sum of \(X_n\)'s, where \(1\leq r<\infty\), \(t=\min\{p,r,r'\}\) and \(p=\inf\{p_n\}\). It is shown that the \((p,p')\) Clarkson inequality is preserved by quotient maps and a new proof of the Takahashi-Kato theorem, stating that the \((p,p')\) Clarkson inequality holds in a Banach space \(X\) if and only if it holds in its dual space \(X^*\), is given.