Differentiability on semilinear spaces

A new differentiability theory for mappings between semilinear spaces is introduced in this work. The main aim is to develop a solid analysis framework that may cover any type of mapping in the semilinear framework and not only curves with real domain as the majority of recent methods in this area do. As a result, a convenient framework is provided within which problems that go beyond the borders of classical analysis and differential geometry for single-valued mappings can be studied. Such issues could include stochastic manifolds, set valued and fuzzy differential equations, etc. On the other hand, the proposed environment keeps all the characteristics of a differentiability theory, being a natural generalization of the most widely used differentiation theories both for single-valued and set valued functions.