- [Banach Space Introduction](../../MATH%20601%20Functional%20Analysis,%20Measure%20Theory/Functional%20Spaces/Banach%20Space%20Introduction.md) - [Linear Functionals and Dual Spaces](../../MATH%20601%20Functional%20Analysis,%20Measure%20Theory/Linear%20Functionals%20and%20Dual%20Spaces.md) - [Hilbert Space Introduction](../../MATH%20601%20Functional%20Analysis,%20Measure%20Theory/Functional%20Spaces/Hilbert%20Space%20Introduction.md) --- ### **Intro** We introduce the idea of Frechet differentiability under the context of Frechet Derivative. #### **Def | Banach Frechet Derivatives** > Let $U, V$ be Banach spaces with their norm. Let $f: W \mapsto V$ where $W$ is an open subset of $U$ then the Frechet Derivative of the function at point $\bar x$ is a linear mapping $A$ such that > $ > \begin{aligned} > \lim_{x\rightarrow \bar x} > \frac{\Vert f(x) - f(\bar x) - A(x - \bar x)\Vert_U}{\Vert x - \bar x\Vert_V} = 0. > \end{aligned} > $ > The linear function $A$ maps from $W$ to $V$. **Remarks** It requires some thoughts to show that it's unique. It requires more thoughts to say that $A$ is a bounded linear mapping between the Banach Spaces.