Rather than a multilinear function, this is instead a homogeneous function of degree in There is another candidate for the definition of the higher order derivative, the function that arises naturally in the calculus of variations as the second variation of at least in the special case where is scalar-valued. However, this may fail to have any reasonable properties at all, aside from being separately homogeneous in and It is desirable to have sufficient conditions in place to ensure that is a symmetric bilinear function of and and that it agrees with the polarization of For instance, the following sufficient condition holds . Suppose that is in the sense that the mapping is continuous in the product topology, and moreover that the second derivative defined by () is also continuous in the sense that is continuous. Then is bilinear and symmetric in and By virtue of the bilinearity, the polarization identity holds relating the second order derivative with the differential Similar conclusions hold for higher order derivatives. Properties A version of the fundamental theorem of calculus holds for the Gateaux derivative of provided is assumed to be sufficiently continuously differentiable. Specifically: Suppose that is in the sense that the Gateaux derivative is a continuous function Then for any and where the integral is the Gelfand–Pettis integral (the weak integral) (). Many of the other familiar properties of the derivative follow from this, such as multilinearity and commutativity of the higher-order derivatives. Further properties, also consequences of the fundamental theorem, include: (The chain rule) for all and (Importantly, as with simple partial derivatives, the Gateaux derivative does satisfy the chain rule if the derivative is permitted to be discontinuous.) (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 Example Let be the Hilbert space of square-integrable functions on a Lebesgue measurable set in the Euclidean space The functional where is a real-valued function of a real variable and is defined on with real values, has Gateaux derivative Indeed, the above is the limit of See also References . . . . . . Generalizations of the derivative Topological vector spaces