Differential calculus founded on an isomorphism (Q690377)

The paper presents a new approach to calculus theory via categorical techniques. The starting point is a natural isometric linear isomorphism (called Foundational Isomorphism). For a nondegenerate compact interval \(I\) and a real Banach space \(E\), \(C(I\times I,E)\) denotes the Banach space of all \(E\)-valued continuous maps defined on \(I\times I\) and \(\text{ad }C(I\times I,E)\) denotes the Banach subspace of all ``additive'' \(A\) in \(C(I\times I,E)\) for which: \[ (\beta-\alpha)A(\alpha,\beta) + (\gamma-\beta)A(\beta,\gamma) + (\alpha-\gamma)A(\gamma-\alpha) = 0,\quad (\alpha,\beta,\gamma \in I). \] The mentioned isomorphism \(\text{ed}^ I\) acts in the category of real Banach spaces Ban, with components \[ \text{ed}^ I_ E: \text{ad }C(I\times I,E) \to C(I,E),\quad \text{ed}(A)(\lambda) = A(\lambda,\lambda). \] Moreover, every component is isometric. The case of noncompact interval is also treated, and several calculus properties (including the fundamental theorem of calculus) are obtained as direct corollaries.