In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics. Unlike other forms of derivatives, the Gateaux differential of a function may be nonlinear. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as , draw a further distinction between the Gateaux differential (which may be nonlinear) and the Gateaux derivative (which they take to be linear). In most applications, continuous linearity follows from some more primitive condition which is natural to the particular setting, such as imposing complex differentiability in the context of infinite dimensional holomorphy or continuous differentiability in nonlinear analysis. Definition Suppose and are locally convex topological vector spaces (for example, Banach spaces), is open, and The Gateaux differential of at in the direction is defined as If the limit exists for all then one says that is Gateaux differentiable at The limit appearing in () is taken relative to the topology of If and are real topological vector spaces, then the limit is taken for real On the other hand, if and are complex topological vector spaces, then the limit above is usually taken as in the complex plane as in the definition of complex differentiability. In some cases, a weak limit is taken instead of a strong limit, which leads to the notion of a weak Gateaux derivative. Linearity and continuity At each point the Gateaux differential defines a function This function is homogeneous in the sense that for all scalars However, this function need not be additive, so that the Gateaux differential may fail to be linear, unlike the Fréchet derivative. Even if linear, it may fail to depend continuously on if and are infinite dimensional. Furthermore, for Gateaux differentials that linear and continuous in there are several inequivalent ways to formulate their continuous differentiability. For example, consider the real-valued function of two real variables defined by This is Gateaux differentiable at with its differential there being However this is continuous but not linear in the