The provocative title may be supposed to hold if the axiom of choice is weakened. Even more: The dual space of any Köthe space (real-valued and over a σ-finite measure space) coincides with its associate space in the canonical way; in particular, any Köthe space with Fatou's property is re-flexive. We will show that all this is true in ZF + DC + PM ω , where PM ω is introduced in the text and holds e.g. in Solovay's model.