Spaceability of sets in \(L^{p} \times L^{q}\) and \(C_{0}\times C_{0}\) (Q289504)

A subset \(E\) of a topological vector space \(X\) is called spaceable if \(E\cup \{0\}\) contains an infinitely dimensional closed subspace of \(X\). Let \((X,\Sigma,\mu)\) be a measure space and \(p,q,r>0\). The authors give conditions on \(p,q,r\) under which \(E=\{(f,g)\in L^p(\mu)\times L^q(\mu): fg\notin L^r(\mu)\}\) is spaceable in \(L^p(\mu)\times L^q(\mu)\). Let \(\mu\) be a regular Borel measure on a locally compact space \(X\) and \(C_0(X)\) the space of all real-valued continuous functions on \(X\) vanishing at infinity. It is characterized when \(E=\{(f,g)\in C_0(X)\times C_0(X): fg\notin L^1(\mu)\}\) is spaceable in \(C_0(X)\times C_0(X)\). Let \(\mu\) be a Haar measure on a locally compact group. The authors study spaceable subsets \(E\) of \(L^p(\mu)\times L^q(\mu)\) using a condition on the convolution \(f\ast g\) for \((f,g)\in E\).