Polynomials on Banach Spaces: Zeros and Maximal Points

Norms of polynomials and capacities on Banach spaces

“…Notice that, as an application of Lemma 3.2, the result obtained in Theorem 2.3 admits of an equivalent formulation in terms of capacities, namely, r ~(a + ~Bx) -1 + Ilal]" 4 …”

“…The proof of Theorem 2.1 relies on an infinite-dimensional version of Bernstein's inequality due to Harris [3], who stated it only for homogeneous polynomials, although it also works for arbitrary polynomials, as Tonge and Lacruz [4] pointed out. Notice that these definitions also make sense when S is a subset of a complex Banach space X.…”

Where do homogeneous polynomials attain their norm?

“…In recent years there has been considerable interest in the study of geometric properties of the space P( k E). It has been shown not to be strictly convex [13], extremal and smooth points of the unit ball have been characterised for a number of spaces ( [3], [4], [5], [6], [7], [8], [9], [10]), and related problems have been considered, often in the finite-dimensional setting ( [1], [2], [11], [12]).…”