A modal logic internalizing normal proofs
In the proof-theoretic study of logic, the notion of normal proof has been understood and investigated as a metalogical property. Usually we formulate a system of logic, identify a class of proofs as normal proofs, and show that every proof in the system reduces to a corresponding normal proof. This paper develops a system of modal logic that is capable of expressing the notion of normal proof within the system itself, thereby making normal proofs an inherent property of the logic. Using a modality @? to express the existence of a normal proof, the system provides a means for both recognizing and manipulating its own normal proofs. We develop the system as a sequent calculus with the implication connective @? and the modality @?, and prove the cut elimination theorem. From the sequent calculus, we derive two equivalent natural deduction systems.
Let X and Y be two inflnite dimensional real or complex Banach spaces. In this note we determine the forms of surjective additive maps ` : L(X) ! L(Y ) preserving the kernel's dimension or the range's codimension. As consequence, we establish that ` : L(X) ! L(X) preserves the kernel (respectively, the range) if and only if there exists an invertible operator A 2 L(X) such that `(T) = AT (respectively, `(T) = TA) for all T 2 L(X).