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**1 - 2**of**2**### Cartesian Products of Family of Real Linear Spaces

"... Summary. In this article we introduced the isomorphism mapping between cartesian products of family of linear spaces [4]. Those products had been formalized by two different ways, i.e., the way using the functor [:X,Y:] and ones using the functor “product”. By the same way, the isomorphism mapping w ..."

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Summary. In this article we introduced the isomorphism mapping between cartesian products of family of linear spaces [4]. Those products had been formalized by two different ways, i.e., the way using the functor [:X,Y:] and ones using the functor “product”. By the same way, the isomorphism mapping was defined between Cartesian products of family of linear normed spaces also.

### DOI: 10.2478/v10037-012-0005-1 versita.com/fm/ Differentiable Functions on Normed Linear

"... Summary. 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-val ..."

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Summary. 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 ||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].