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eISSN 0454-8124
pISSN 1225-6951

### Published Online

Kyungpook Mathematical Journal

Published online July 14, 2020

### Extreme Points, Exposed Points and Smooth Points of the Space $L s( 2𝑙∞3)$

Sung Guen Kim

Department of Mathematics, Kyungpook National University, Daegu 41566, Repub- lic of Korea
e-mail : sgk317@knu.ac.kr

Received: September 5, 2019; Revised: March 10, 2020; Accepted: April 21, 2020

### Abstract

We present a complete description of all the extreme points of the unit ball of $L s( 2𝑙∞3)$ which leads to a complete formula for $∥f ∥$ for every $𝑓∈ L s( 2𝑙∞3)*$. We also show that $extB L s( 2𝑙∞3)⊂ extB L s( 2𝑙∞n)$ for every n ≥ 4. Using the formula for $∥f ∥$ for every $f∈ L s( 2𝑙∞3)*$, we show that every extreme point of the unit ball of $L s( 2𝑙∞3)$ is exposed. We also characterize all the smooth points of the unit ball of $L s( 2𝑙∞3)$.

Keywords: symmetric bilinear forms on ℝ,3 with the supremum norm, extreme points, exposed points, smooth points