We write B_E for the closed unit ball of a real Banach space E and the dual space of E is denoted by E^*. x ∈ B_E is called an extreme point of B_E if y, z ∈ B_E with x=12(y+z) implies x = y = z. x ∈ B_E is called a smooth point of B_E if there is a unique f ∈ E^* so that f(x) = 1 = ||f||. We denote by extB_E and smB_E the sets of extreme and smooth points of B_E, respectively. A mapping P : E → ℝ is a continuous 2-homogeneous polynomial if there exists a continuous symmetric bilinear form L on the product E×E such that P(x) = L(x, x) for every x ∈ E. We denote by ℒ_s(²E) the Banach space of all continuous symmetric bilinear forms on E endowed with the norm ||L|| = sup_{||x||=||y||=1} |L(x, y)|. ℘(²E) denotes the Banach space of all continuous 2-homogeneous polynomials from E into ℝ endowed with the norm ||P|| = sup_||x||=1 |P(x)|. For more details about the theory of polynomials on a Banach space, we refer to [7].

In 1998, Choi and the author [3] characterized the smooth points of the unit ball of P(l222) and in 1999, Choi and the author [5] characterized the smooth points of the unit ball of P(l212) and studied smooth polynomials of ℘(²l₁). In 2009, the author [10] classified the smooth symmetric bilinear forms of Ls(l2∞2). We refer to ([1], [3–6], [8–19] and references therein) for some recent work about extremal properties of multilinear mappings and homogeneous polynomials on some classical Banach spaces. Let 0 < w < 1 be fixed. We denote the two dimensional real predual of Lorentz sequence space by

In fact, d_*(1, w)² = ℝ² with the octagonal norm of weight w. We will denote by T((x₁, x₂), (y₁, y₂)) = ax₁x₂ +by₁y₂ +c(x₁y₂ +x₂y₁) a symmetric bilinear form on d_*(1, w)². Recently, the author [12] computed the norm of T ∈ ℒ_s(²d_*(1, w)²) in terms of their real coefficients and determined all the extreme symmetric bilinear forms of the unit ball of ℒ_s(²d_*(1, w)²). In this paper, using results of the previous work [12], we classify the smooth symmetric bilinear forms of the unit ball of the space ℒ_s(²d_*(1, w)²). We also show that the unit sphere S_{ℒs(²d*(1,w)²)} is the disjoint union of the sets of smooth points, extreme points and the set A as follows:

where A consists of ax₁x₂ + by₁y₂ + c(x₁y₂ + x₂y₁) with (a = b = 0, c=±11+w2), (a ≠ b, ab ≥ 0, c = 0), (a = b, 0 < ac, 0 < |c| < |a|), (a ≠ |c|, a = −b, 0 < ac, 0 < |c|), (a=1-w1+w,b=0,c=11+w), (a=1+w+w(w2-3)c1+w2,b=w-1+(1-3w2)cw(1+w2),12+2w<c<1(1+w)2(1-w),c≠11+2w-w2), (a=1+w(1+w)c1+w,b=-1+(1+w)cw(1+w),0<c<12+2w) or ( a=1-w(1+w)c1+w,b=1-(1+w)c1+w,11+w<c<1(1+ω)2(1-ω)).

Let T((x₁, y₁), (x₂, y₂)) = ax₁x₂ + by₁y₂ + c(x₁y₂ + x₂y₁) ∈ ℒ_s(²d_*(1, w)²) for some reals a, b, c. By substituting ((x₁, y₁), (x₂, y₂)) in T for ((x₁, y₁), (−x₂, −y₂)) or ((x₁, −y₁), (x₂, −y₂)) or ((y₁, x₁), (y₂, x₂)), we may assume that |b| ≤ a, c ≥ 0.

Theorem 2.1.([12, Theorem 2.1])

Let T((x₁, y₁), (x₂, y₂)) = ax₁x₂ + by₁y₂ + c(x₁y₂ + x₂y₁) ∈ ℒ_s(²d_*(1, w)²) with |b| ≤ a, c ≥ 0. Then

In fact, we have the following:

Case 1: b ≥ 0

Subcase 1: c > a

• If w≤c-ac-b, then ||T|| = (a + b)w + c(1 + w²).

• If w>c-ac-b, then ||T|| = bw² + 2cw + a.

Subcase 2: If c ≤ a, ||T|| = bw² + 2cw + a.

Case 2: b < 0

Subcase 1: c < |b|

• If w≤c∣b∣, then ||T|| = max{bw² + 2cw + a, (a − b)w + c(1 − w²)}.

• If w>c∣b∣, then ||T|| = max{a − bw², (a − b)w + c(1 − w²)}.

Subcase 2: c ≥ |b|

• If w≤∣b∣c, then ||T|| = max{bw² + 2cw + a, (a − b)w + c(1 − w²)}.

• If w>∣b∣c, then ||T|| = max{bw² + 2cw + a, (a + b)w + c(1 + w²)}.

By Theorem 2.1, if ||T|| = 1, then |a| ≤ 1, |b| ≤ 1, ∣c∣ ≤11+w2.

Theorem 2.2

Let T((x₁, y₁), (x₂, y₂)) = ax₁x₂ + by₁y₂ + c(x₁y₂ + x₂y₁) ∈ ℒ_s(²d_*(1, w)²). Then the following are equivalent:

• ax₁x₂ + by₁y₂ + c(x₁y₂ + x₂y₁) is a smooth point of ℒ_s(²d_*(1, w)²);

• − (ax₁x₂ + by₁y₂ + c(x₁y₂ + x₂y₁)) is a smooth point of ℒ_s(²d_*(1, w)²);

• ax₁x₂ + by₁y₂ − c(x₁y₂ + x₂y₁) is a smooth point of ℒ_s(²d_*(1, w)²);

• bx₁x₂ + ay₁y₂ + c(x₁y₂ + x₂y₁) is a smooth point of ℒ_s(²d_*(1, w)²).

Theorem 2.3.([12, Theorem 2.3])

Let T((x₁, y₁), (x₂, y₂)) = ax₁x₂ + by₁y₂ + c(x₁y₂ + x₂y₁) ∈ ℒ_s(²d_*(1, w)²). Then

• Let w<2-1. Then T is an extreme point of ℒ_s(²d_*(1, w)²) if and only ifT∈{±x1x2,±y1y2,±11+w2(x1x2+y1y2),±1(1+w)2[x1x2+y1y2±(x1y2+x2y1)],±11+2w-w2[x1x2-y1y2±(x1y2+x2y1)],±11+w2[x1x2-y1y2±w(x1y2+x2y1),±11+w2[wx1x2-wy1y2±(x1y2+x2y1)],±1(1+w)2(1-w)[(1-w-w2)x1x2-wy1y2±(x1y2+x2y1)],±1(1+w)2(1-w)[wx1x2-(1-w-w2)y1y2±(x1y2+x2y1)]}.

• Let w<2-1. Then T is an extreme point of ℒ_s(²d_*(1, w)²) if and only ifT∈{±x1x2,±y1y2,±2+24(x1x2+y1y2),±12[x1x2+y1y2±(x1y2+x2y1)],±24[x1x2+y1y2±(2+1)(x1y2+x2y1)],±24[(2+1)(x1y2-x2y1)±(x1y2+x2y1)]}.

• Let w<2-1. Then T is an extreme point of ℒ_s(²d_*(1, w)²) if and only ifT∈{±x1x2,±y1y2,±11+w2(x1x2+y1y2),±1(1+w)2[x1x2+y1y2±(x1y2+x2y1)],±11+2w-w2[x1x2-y1y2±(x1y2+x2y1)],±11+w2[x1x2-y1y2±1-w1+w(x1y2+x2y1)],±11+w2[1-w1+w(x1x2-y1y2)±(x1y2+x2y1)],±12+2w[(2+w)x1x2-1wy1y2±(x1y2+x2y1)],±12+2w[1wx1x2-(2+w)y1y2±(x1y2+x2y1)]}.

Theorem 2.4

Let f ∈ ℒ_s(²d_*(1, w)²)^*and α = f(x₁x₂), β = f(y₁y₂), γ = f(x₁y₂ + x₂y₁).

• Let w<2-1. Then‖f‖ =max{∣α∣,∣β∣,11+w2∣α+β∣,1(1+w)2(∣α+β∣+∣γ∣),11+2w-w2(∣α-β∣+∣γ∣),11+w2(∣α-β∣+ w∣γ∣),11+w2(w∣α-β∣+∣γ∣),1(1+w)2(1-w)(∣(1-w-w2)α-wβ∣+∣γ∣),1(1+w)2(1-w)(∣wα-(1-w-w2)β∣+∣γ∣)}.

• Let w<2-1. Then‖f‖ =max{∣α∣,∣β∣,2+24∣α+β∣,12(∣α+β∣+∣γ∣),24(∣α-β∣+ (2+1)∣γ∣),24((2+1)∣α-β∣+∣γ∣)}.

• Let 2-1<w. Then‖f‖ =max{∣α∣,∣β∣,11+w2∣α+β∣,1(1+w)2(∣α+β∣+∣γ∣),11+2w-w2(∣α-β∣+∣γ∣,11+w2(∣α-β∣+1-w1+w∣γ∣),11+w2(1-w1+w∣α-β∣+∣γ∣),12+2w(∣(2+w)α-1wβ∣+∣γ∣),12+2w(∣1wα-(2+w)β∣+∣γ∣)}.

Theorem 2.5

Let T((x₁, y₁), (x₂, y₂)) = ax₁x₂ + by₁y₂ + c(x₁y₂ + x₂y₁) ∈ ℒ_s(²d_*(1, w)²) with |b| < a, c > 0. Let S = {bw² + 2cw + a, a − bw², (a + b)w + c(1 + w²), (a − b)w + c(1 − w²)}. Then T ∈ smB_{ℒs(²d*(1,w)²)}if and only if there exists a unique l ∈ S such that l = 1.