The Open Mapping Theorem – that a surjective continuous linear map between Fréchet spaces is open – is one of the cornerstones of linear functional analysis. Effros’ Theorem is a group-action counterpart: a continuous transitive action by a Polish group G on a non-meagre space X is open (i.e. point-evaluation maps $g \mapsto gx$ are open). We discuss action, micro-transitivity and shift-compactness. We investigate what we call the crimping property (arising in Banach’s classic book of 1932). It turns out that this is equivalent to the Effros property. Various related results are proved.

Once again we use the Baire Category Theorem to prove results about linear maps between Banach spaces. We prove the Open Mapping Theorem and, as a corollary, the Inverse Mapping Theorem, which allows for some simplification in the spectral theory of bounded operators. As an application, we prove the existence of a ‘basis constant’ for any Schauder basis in a separable Banach space. Finally, we use the Inverse Mapping Theorem to prove the Closed Graph Theorem, which gives an alternative way to check whether a linear map T from X into Y is bounded, provided both X and Y are Banach spaces.