Gâteaux gradient

of a functional $ f $ at a point $ x _ {0} $ of a Hilbert space $ H $

The vector in $ H $ equal to the Gâteaux derivative $ f _ {G} ^ { \prime } ( x _ {0} ) $ of $ f $ at $ x _ {0} $. In other words, the Gâteaux gradient is defined by the formula

$$ f ( x _ {0} + h) = \ f ( x _ {0} ) + ( f _ {G} ^ { \prime } ( x _ {0} ), h) + \epsilon ( h), $$

where $ \epsilon ( th)/t \rightarrow 0 $ as $ t \rightarrow 0 $. In an $ n $- dimensional Euclidean space the Gâteaux gradient $ f _ {G} ^ { \prime } ( x _ {0} ) $ is the vector with coordinates

$$ \left ( \frac{\partial f ( x _ {0} ) }{\partial x _ {1} } \dots \frac{\partial f ( x _ {0} ) }{\partial x _ {n} } \right ) , $$

and is simply known as the gradient. The concept of the Gâteaux gradient may be extended to the case when $ X $ is a Riemannian manifold (finite-dimensional) or an infinite-dimensional Hilbert manifold and $ f $ is a smooth real function on $ X $. The growth of $ f $ in the direction of its Gâteaux gradient is larger than in any other direction passing through the point $ x _ {0} $.