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Norm attaining bilinear forms on the plane with the octagonal norm

Kim Sung Guen

Commentationes Mathematicae

2023

Vol. 61, No. 1/2

15--24

For given unit vectors x² , ⋯, xn of a real Banach space E, we define NAL(n E)(x² , . . . , xn ) = T ∈ L(n E) ∶ ST(x² , . . . , xn )S = YTY = ², where L(n E) denotes the Banach space of all continuous n-linear forms on E endowed with the norm YTY = sup{ST(x² , . . . , xn )S ∶ Yxk Y = ², ² ⩽ k ⩽ n}. In this paper, we classify NA(L(_ R_ o(w)))((x² , x_ ), (y² , y_ )) for unit vectors (x² , x_ ), (y² , y_ ) ∈ R_ o(w) , where R_ o(w) = R_ with the octagonal norm with weight x < w < ²For given unit vectors x² , ⋯, xn of a real Banach space E, we define NAL(n E)(x² , . . . , xn ) = T ∈ L(n E) ∶ ST(x² , . . . , xn )S = YTY = ², where L(n E) denotes the Banach space of all continuous n-linear forms on E endowed with the norm YTY = sup{ST(x² , . . . , xn )S ∶ Yxk Y = ², ² ⩽ k ⩽ n}. In this paper, we classify NA(L(_ R_ o(w)))((x² , x_ ), (y² , y_ )) for unit vectors (x² , x_ ), (y² , y_ ) ∈ R_ o(w) , where R_ o(w) = R_ with the octagonal norm with weight x < w < ².