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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 $x _ {0}$.

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