The Open Mapping Theorem

The open mapping theorem (or the Banach-Schauder theorem, if you prefer) is an incredibly important, relatively straightforward and digestible result in functional analysis which plays a crucial role in a large variety of other interesting theorems.  As an exercise, we’ll prove the open mapping theorem here in the standard fashion.  This will hopefully serve as a useful reference for later posts about metric regularity and linear openness, which serve as a measuring device to quantify differing degrees of linear openness.