The open mapping theorem is usually proved in most texts using Baires Category theorem which depends upon the axiom of choice.

But if one studies differential calculus in Banach spaces say as in Dieuodenne Foundations of Modern Analysis the theorem is the first part of Inverse mapping theorem( as proved in Walter Rudin's classic Principles of Mathematical Analysis and the proof carries over to Banach space setting ) as a continuous linear map is differentiable This proof does not depend upon Baires Category Theorem.


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