open mapping theorem
The open mapping theorem says that a bounded linear operator between Banach spaces that hits everything cannot squash space — it must spread open regions onto open regions. Intuitively, if the operator reaches every target then it cannot be infinitely compressive anywhere; surjectivity alone forces a quantitative openness. There is no slow, asymptotic way to cover the whole target.
Precisely: let X and Y be Banach spaces and T : X -> Y a bounded linear operator that is surjective. Then T is an open map — the image T(U) of every open set U is open in Y. Equivalently, there is a constant c > 0 such that every y in Y has a preimage x with ||x|| <= (1/c) ||y||: targets of bounded size are reached from sources of bounded size.
The headline corollary is the bounded inverse theorem: if T is a bounded linear bijection between Banach spaces, then T inverse is automatically bounded too, so T is a topological isomorphism. Completeness of both spaces is indispensable — the proof rests on the Baire category theorem — and the conclusion genuinely fails for incomplete spaces or for nonsurjective operators. It is a striking instance of qualitative hypotheses (linear, bounded, onto, complete) forcing a quantitative conclusion.
Suppose a single vector space carries two norms ||·||_a and ||·||_b, complete under each, with ||x||_b <= C ||x||_a for all x. The identity map (X, ||·||_a) -> (X, ||·||_b) is then a bounded bijection, so by the bounded inverse theorem there is also c > 0 with ||x||_a <= (1/c) ||x||_b: a one-sided comparison of two complete norms automatically upgrades to equivalence.
Two complete norms comparable in one direction are automatically equivalent.
All three pillars — open mapping, closed graph, uniform boundedness — descend from the Baire category theorem and so demand completeness. Take away the assumption that both spaces are Banach and each statement collapses; this is the single most important caveat to keep in mind when invoking them.