A Fréchet space \(E\) is called primary if whenever \(E= F\oplus G\), then either \(F\) or \(G\) is isomorphic to \(E\). It is known that \(\omega\), \((c_0)^{\mathbb{N}}\) and \((\ell^p)^{\mathbb{N}}\), with \(1\leq p\leq \infty\), are primary Fréchet spaces without continuous norm. However, it is unknown whether there are other countable products of Banach spaces which are primary.
It is the purpose of this paper to prove that the spaces \(X^{\mathbb{N}}\), with \(X\) a separable rearrangement invariant function space on \([0,1]\) with Boyd indices \(1<p_X\) and \(q_X<\infty\), are primary. The considered class of spaces contains many natural spaces like \(L^{\text{loc}}_p(\mathbb{R})\), with \(1<p<\infty\), and the analogous spaces \(L^{\text{loc}}_M(\mathbb{R})\) of Orlicz-type.