## Differentiable Functions on Normed Linear Spaces

In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If ||ƒ'(*x* + *t* · *h*)|| is bounded for *t* between 0 and 1 by some constant *M*, then ||ƒ(*x* + *t* · *h*) - ƒ(*x*)|| ≤ *M* · ||*h*||. This theorem is called the mean value theorem for vector-valued functions. By this theorem, the relation between the (total) derivative and the partial derivatives of a function is derived [23].