A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ
We construct two examples of infinite spaces $X$ such that there is no continuous linear surjection from the space of continuous functions $c_p(X)$ onto $c_p(X) × ℝ$. In particular, $c_p(X)$ is not linearly homeomorphic to $c_p(X) × ℝ$. One of these examples is compact. This answers some questions of Arkhangel'skiĭ.