Journal of the
Korean Mathematical Society JKMS

For every $n\geq 2$, let $\mathbb{R}^n_{\|\cdot\|}$ be $\mathbb{R}^n$ with a norm $\|\cdot\|$ such that its unit ball has finitely many extreme points more than $2n$. We devote to the description of the sets of extreme and exposed points of the closed unit balls of ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ and ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$, where ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ is the space of bilinear forms on $\mathbb{R}^n_{\|\cdot\|}$, and ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$ is the subspace of ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ consisting of symmetric bilinear forms. Let ${\mathcal F}={\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ or ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$. First we classify the extreme and exposed points of the closed unit ball of ${\mathcal F}$. We also show that every extreme point of the closed unit ball of ${\mathcal F}$ is exposed. It is shown that ${ext}B_{{\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})}={ext}B_{{\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})}\cap {\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$ and ${exp}B_{{\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})}={exp}B_{{\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})}\cap {\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$, which expand some results of \cite{18, 23, 28, 29, 35, 38, 40, 41, 43}.

Keywords: Bilinear forms, symmetric bilinear forms, extreme points, exposed points

MSC numbers: 46A22