what is meanning frechet and gateaux derivatives
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I need additional sources to tell the difference between them and how to calculate it
The Fréchet and Gâteaux derivatives are concepts in mathematical analysis and functional analysis that are used to study the differentiation of functions defined on a normed vector space.
The Fréchet derivative is defined for functions between normed spaces. Let's say we have a function f: X -> Y, where X and Y are normed vector spaces. The Fréchet derivative of f at a point x0 in X is a linear operator from X to Y, denoted by Df(x0). It measures the sensitivity of the function f to changes in its input at the point x0.
To find the Fréchet derivative, we need to consider the limit of the difference quotient as it approaches zero. The Fréchet derivative Df(x0) exists if this limit exists and is linear (i.e., satisfies the properties of linearity).
On the other hand, the Gâteaux derivative is defined for functions between Banach spaces. Similar to the Fréchet derivative, we have a function f: X -> Y, but now X and Y are Banach spaces. The Gâteaux derivative of f at a point x0 in X is a linear functional from X to Y, denoted by df(x0, h). It measures the directional derivative of f at the point x0, in the direction of a vector h in X.
To find the Gâteaux derivative, we evaluate the difference quotient of f at the point x0 in the direction of the vector h. This difference quotient should approach the Gâteaux derivative df(x0, h) as the difference h approaches zero.
In summary, the Fréchet and Gâteaux derivatives are tools used to study the differentiation of functions defined on normed or Banach spaces. The Fréchet derivative is a linear operator, while the Gâteaux derivative is a linear functional. Both provide information about the sensitivity or directional derivatives of a function at a given point. To compute these derivatives, we need to evaluate difference quotients and check their limit.