Extreme points of ${\mathcal L}_s(^2l_{\infty})$ and ${\mathcal P}(^2l_{\infty})$

Abstract

For $n\geq 2,$ we show that every extreme point of the unit ball of ${\mathcal L}_s(^2l_{\infty}^n)$ is extreme in ${\mathcal L}_s(^2l_{\infty}^{n+1})$, which answers the question in [Period. Math. Hungar. 2018, 77 (2), 274-290]. As a corollary we show that every extreme point of the unit ball of ${\mathcal L}_s(^2l_{\infty}^n)$ is extreme in ${\mathcal L}_s(^2l_{\infty})$. We also show that every extreme point of the unit ball of ${\mathcal P}(^2l_{\infty}^2)$ is extreme in ${\mathcal P}(^2l_{\infty}^n).$ As a corollary we show that every extreme point of the unit ball of ${\mathcal P}(^2l_{\infty}^2)$ is extreme in ${\mathcal P}(^2l_{\infty})$.

Keywords

• extreme point
• symmetric bilinear form
• 2-homogeneous polynomials on $l_{\infty}$