Differentiable Functions on Normed Linear Spaces1

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 vectorvalued functions is obtained as follows: If ||f ′(x+ t ·h)|| is bounded for t between 0 and 1 by some constant M, then ||f(x+t ·h)−f(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].