h . FA ] 1 0 Ja n 19 95 On the extension of 2 - polynomials Pei

Let X be a normed linear space over K (R or C). A function P : X → K is said to be a 2-polynomial if there is a bilinear functional Π : X × X → K such that P (x) = Π(x, x) for every x ∈ X. The norm of P is defined by P = sup{|P (x)| : x = 1}. It is known that if X is an inner product space, then every 2-polynomial defined in a linear subspace of X can be extended to X preserving the norm. On the other hand, there is a 2-polynomial P defined in a two dimensional subspace of ℓ 3 ∞ such that every extension of P to ℓ 3 ∞ has norm greater than P , (see [1, 2]). Recently, Benítez and Otero [2] showed that if X is a three dimensional real Banach space X such that the unit ball of X is an intersection of two ellipsoids, then every 2-polynomial defined in a linear subspace of X can be extended to X preserving the norm. It is natural to ask [2] Question 1 Suppose X is a norm space such that the unit ball of X is an intersection of two ellipsoids. Can every 2-polynomial defined in a linear subspace of X be extended to X preserving the norm? In this article, we show the answer is affirmative when X is a finite dimensional space. First, we recall the following result in [2].