Norm attaining polynomials

If L ≠ 0 is a continuous symmetric n ‐linear form on a Banach space and L ^ is the associated continuous n ‐homogeneous polynomial, the ratio ∥ L ∥ / ∥ L | | always lies between 1 and n n / n !. At one extreme, if L is defined on Hilbert space, then∥ L ∥ / ∥ L ^ ∥ = 1 . If L attains norm on Hilbert space, then L ^ also attains norm; in this case, we give an explicit construction to provide a unit vector x 0 with∥ L ^ ∥ = | L ^ ( x 0 ) | = ∥ L | . At the other extreme, if∥ L ∥ / ∥ L ^ ∥ = n n / n ! and L attains norm, then L ^ attains norm. We prove that in general the converse is not true.