Article
Kyungpook Mathematical Journal 2018; 58(1): 47-54
Published online March 23, 2018
Copyright © Kyungpook Mathematical Journal.
The Geometry of
L
(
l
2
∞
2
)
Sung Guen Kim
Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea, e-mail:
Received: October 8, 2015; Accepted: January 18, 2016
Abstract
We classify the extreme, exposed and smooth bilinear forms of the unit ball of the space of bilinear forms on
Keywords: Bilinear forms, extreme points, exposed points, smooth points
1. Introduction
We write
In 1998, Choi
In 2009, the author [13] classified the extreme, exposed and smooth points of the unit ball of
We refer to ([1]–[7], [9]–[25] and references therein) for some recent work about extremal properties of multilinear mappings and homogeneous polynomials on some classical Banach spaces. In this paper, we classify the extreme, exposed and smooth bilinear forms of the unit ball of
2. The extreme points of the unit ball of L ( l 2 ∞ 2 )
Let
Then ||
Theorem 2.1
Since {(1, 1), (1,−1), (−1, 1), (−1,−1)} is the set of all extreme points of the unit ball of
Note that if ||
Theorem 2.2
(1)
T is extreme; (2) (
b ,a ,d ,c )is extreme; (3) (
a ,−b ,−c ,d )is extreme; (4) (
c ,d ,a ,b )is extreme; (5) (
c ,d ,b ,a )is extreme.
It follows from Theorem 2.1 and the above remark of Theorem 2.1.
Let
We call
Theorem 2.3
Suppose that
(⇐): Obviously,
Hence, 1 = |
(⇒): Suppose that
Then
Without loss of generality we may assume that
Case 1:
Let
Case 2:
Let
Case 3:
Let
Case 4:
Let
Theorem 2.4
It follows from the proof of Theorem 2.3.
3. The exposed points of the unit ball of L ( l 2 ∞ 2 )
Theorem 3.1
It follows from Theorem 2.3 and the fact that
Note that if ||
Theorem 3.2
([18, Theorem 2.3]).
Theorem 3.3
(1)
T is exposed; (2) (
b ,a ,d ,c )is exposed; (3) (
a ,−b ,−c ,d )is exposed; (4) (
c ,d ,a ,b )is exposed; (5) (
c ,d ,b ,a )is exposed.
It follows from Theorem 2.1 and the above remark of Theorem 2.1.
Now we are in position to describe all the exposed points of the unit ball of
Theorem 3.4. e x p B L ( l 2 ∞ 2 ) = e x t B L ( l 2 ∞ 2 )
It is enough to show that
Claim:
Let
Claim:
Let
4. The smooth points of the unit ball of L ( l 2 ∞ 2 )
Theorem 4.1
(1)
T is smooth; (2) (
b ,a ,d ,c )is smooth; (3) (
a ,−b ,−c ,d )is smooth; (4) (
c ,d ,a ,b )is smooth; (5) (
c ,d ,b ,a )is smooth.
It follows from Theorem 2.1 and the above remark of Theorem 2.1.
Theorem 4.2
By Theorem 4.1, we may assume that
(⇒): Suppose that
Obviously,
Hence, if
(⇐): Let
Case 1: 0 < |
By Theorem 2.1, ||
By Theorem 2.1, it follows that, for a sufficiently large
From (
Case 2: 0 < |
By Case 1 and Theorem 4.1,
Remark
Nine months after the acceptance of this paper in Kyungpook Mathematical Journal, the author found that W. Cavalcante and D. Pellegrino had proved in [2], independently, Theorems 2.1, 2.4 and 3.4.
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