In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is Gateaux derivative between Fréchet spaces, is significantly weaker than the derivative in a Banach space, even between general topological vector spaces. Nevertheless, it is the weakest notion of differentiation for which many of the familiar theorems from calculus hold. In particular, the chain rule is true. With some additional constraints on the Fréchet spaces and functions involved, there is an analog of the inverse function theorem called the Nash–Moser inverse function theorem, having wide applications in nonlinear analysis and differential geometry. Mathematical details Formally, the definition of differentiation is identical to the Gateaux derivative. Specifically, let and be Fréchet spaces, be an open set, and be a function. The directional derivative of in the direction is defined by if the limit exists. One says that is continuously differentiable, or if the limit exists for all and the mapping is a continuous map. Higher order derivatives are defined inductively via A function is said to be if It is or smooth if it is for every Properties Let and be Fréchet spaces. Suppose that is an open subset of is an open subset of and are a pair of functions. Then the following properties hold: Fundamental theorem of calculus. If the line segment from to lies entirely within then The chain rule. For all and Linearity. is linear in More generally, if is then is multilinear in the 's. Taylor's theorem with remainder. Suppose that the line segment between and lies entirely within If is then where the remainder term is given by Commutativity of directional derivatives. If is then for every permutation σ of The proofs of many of these properties rely fundamentally on the fact that it is possible to define the Riemann integral of continuous curves in a Fréchet space. Smooth mappings Surprisingly, a mapping between open subset of Fréchet spaces is smooth (infinitely often differentiable) if it maps smooth curves to smooth curves; see Convenient analysis. Moreover, smooth curves in spaces of smooth functions are just smooth functions of one variable more. Consequences in differential geometry The existence of a chain rule allows for the definition of a manifold modeled on a Frèchet space: a Fréchet manifold. Furthermore, the linearity of the derivative implies that there