A Note on the Existence of Real Two-dimensional Symmetric Subspaces of \(L^p[-1,1]\)

Abstract

An \(n\)-dimensional normed space is said to be symmetric if there exists a basis \(v_1, \dots, v_n\) such that \[ \left\| \sum_{i} |c_i| v_i \right\| = \left\| \sum_i c_{\sigma(i) } v_i \right\| \] for all \(c_1, \dots, c_n \in \mathbb R\) and for every \(\sigma \in \text{Perm}(n)\), the set of all permutations on \(n\) elements. In this paper we 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, we organize our consideration by (so-called) diagonal, triangular and symmetric bases. We find that, outside the \(p=4\) and \(p=2\) cases, symmetry is not possible for the bases considered.