FréchetDerivative

The Fréchet derivative is a generalization of the concept of a derivative to Banach spaces, which are complete normed vector spaces. In calculus, the derivative of a function $f: \mathbb{R} \to \mathbb{R}$ at a point $x$ is a number that gives the best linear approximation to $f$ near $x$. The Fréchet derivative extends this idea to functions between Banach spaces, capturing the essence of differentiability in a more abstract setting.

Let $X$ and $Y$ be Banach spaces, and let $f: X \to Y$ be a function. We say that $f$ is Fréchet differentiable at a point $x_0 \in X$ if there exists a bounded linear operator $A: X \to Y$ such that

$$ \lim_{{h \to 0}} \frac{| f(x_0 + h) - f(x_0) - A(h) |_Y}{| h |_X} = 0. $$

Here, $| \cdot |_X$ and $| \cdot |_Y$ denote the norms on $X$ and $Y$, respectively.

In this definition, the operator $A$ serves as the "best linear approximation" to $f$ near $x_0$, analogous to the role of the derivative in ordinary calculus. If such an operator $A$ exists, it is unique and is called the Fréchet derivative of $f$ at $x_0$, denoted by $f'(x_0)$ or $Df(x_0)$.

The Fréchet derivative generalizes several other types of derivatives, including the ordinary derivative for functions $f: \mathbb{R} \to \mathbb{R}$, the gradient for functions $f: \mathbb{R}^n \to \mathbb{R}$, and the Jacobian for functions $f: \mathbb{R}^n \to \mathbb{R}^m$.

The concept of the Fréchet derivative is fundamental in functional analysis, optimization, and the calculus of variations, providing a robust framework for analyzing the behavior of functions between infinite-dimensional spaces.