The norming sets of multilinear forms on a certain normed space R^n
Let n, mN, n, m≥ 2 and E a Banach space. An element (x_1, …, x_n)E^n is called a norming point of TL(^n E) if \|x_1\|=…=\|x_n\|=1 and |T(x_1, …, x_n)|=\|T\|, where L(^n E) denotes the space of all continuous n-linear forms on E. For TL(^n E), we define Norm(T) as the set of all (x_1, …, x_n)E^n which are the norming points of T. Let R^n_\|·\|=R^n with a norm satisfying that {W_1, …, W_n} forms a basis and the set of all extreme points of B_R^n_\|·\| is {± W_1, …, ± W_n}. In the paper we characterize Norm(T) for every TL(^m R^n_\|·\|) as follows: Let T=(T(W_i_1, … W_i_m))_1≤ i_k≤ n,1≤ k≤ m L(^m R^n_\|·\|), \|T\|=1, S_T=(b_i_1… i_m)_1≤ i_k≤ n,1≤ k≤ m L(^m R^n_\|·\|) such that b_i_1… i_m=T(W_i_1, … W_i_m) if |T(W_i_1, … W_i_m)|=1 and b_i_1… i_m=1 if |T(W_i_1, … W_i_m)|<1, and A is the Cartesian product of the set {1, …, n}, M is the set of indices (i_1, …, i_m)A such that |T(W_i_1, … W_i_m)|<1. Then, gather* Norm(T)=_(i_1, …, i_m)M _j=1^m {( _1≤ i≤ ns_i^(1)W_i, …, _1≤ i≤ ns_i^(j-1)W_i, _1≤ i≤ ns_i^(j)W_i-s_i_jW_i_j, _1≤ i≤ ns_i^(j+1)W_i, …, _1≤ i≤ ns_i^(m)W_i) (_1≤ i≤ ns_i^(1)W_i, …, _1≤ i≤ ns_i^(m)W_i)Norm(S_T)}. gather* , number=2, journal=Matematychni Studii, author=Kim, Sung Guen, year=2024, month=Dec., pages=192-198