Open mapping theorem

The open mapping theorem (or Banach-Schauder theorem) states that a surjective continuous linear operator between Banach spaces is an open map — it carries open sets to open sets. This is profoundly false in incomplete normed spaces: a bijective bounded operator can have an unbounded inverse, and the theorem's guarantee of boundedness is precisely what completeness buys. The theorem is the reason that isomorphism in functional analysis means bounded isomorphism with bounded inverse, and it underlies the closed graph theorem, which states that a linear operator with closed graph is continuous. The open mapping theorem is not a curiosity about topology; it is the statement that in complete spaces, algebraic invertibility implies topological invertibility. The proof uses the Baire category theorem, and this connection reveals that completeness is not merely about convergence but about the topological richness of the space.