A note on the existence of real two-dimensional symmetric subspaces of \(L^p[-1,1]\) (Q785266)

An \(n\)-dimensional normed space is said to be symmetric if there exists a basis \(v_1,v_2,\dots,v_n\) such that \( \| \sum_i |c_i|v_i\|=\|\sum_i c_{\sigma(i)}v_i\| \) for all \(c_1,\dots,c_n\in R\) and for every \(\sigma \in \mathrm{Perm}(n)\), the set of all permutations on \(n\) elements. The authors consider the existence of 2-dimensional symmetric subspaces of \(L^p[-1,1]\) spanned by \(t^{k_e}\) and \(t^{k_o}\) for even and odd integers \(k_e\) and \(k_o\). As the question of symmetry is basis-specific, the considerations are organized by (suitably defined) diagonal, triangular and symmetric bases. It is proved that outside the \(p = 4\) and \(p = 2\) cases, symmetry is not possible for the bases considered.