A bounded bijective linear map between Banach spaces has a bounded inverse. This follows because an open bijection has continuous inverse.
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The Bounded Inverse Theorem is a result in functional analysis that deals with bounded linear operators between Banach spaces. It provides conditions under which the inverse of a bounded linear operator is also bounded. This theorem is particularly important in the context of linear operators because it helps establish when an operator has a well-defined and continuous (bounded) inverse.