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• # Artykuł - szczegóły

## Formalized Mathematics

2012 | 20 | 1 | 31-40

## Differentiable Functions on Normed Linear Spaces

EN

### Abstrakty

EN
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].

31-40

wydano
2012-01-01
online
2012-09-12

### Twórcy

autor
• Shinshu University, Nagano, Japan

### Bibliografia

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