Soft question alert.

I want to know why to expect the open mapping theorem to be true.

My thoughts: I know that one nice consequence of the OMT could be thought of as the universal property of quotients in the ("isomorphic") category of Banach spaces (a surjective bounded linear map factors through the quotient and becomes an isomorphism of Banach spaces). This is an appealing reason, but I am wondering if there are other useful perspectives with which to think about this theorem.

I have worked through some counterexamples but still am not really able to put my finger on exactly why it openness may fail for surjective bounded linear maps between spaces which are not necessarily Banach.

It is obviously true in the finite case, one circuitious argument being that the exact sequence $0 \to R^k \to R^n \to R^{n-k} \to 0$ splits, and projection maps are open. Can the open mapping theorem be thought of as saying that surjections in the ("isomorphic") category of Banach spaces are like projections, even thought not all exact sequences split (because not all closed subspaces are complemented)?


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