Linearization of a nonlinear map

For a Fréchet differentiable map $F:U\subset E\to Y$ between real normed spaces, its linearization at $u$ is the operator $DF(u)$ . The associated affine approximation is

F(u+h)=F(u)+DF(u)h+R(u,h),\qquad \|R(u,h)\|_Y=o(\|h\|_E).

The Fréchet derivative controls all small increments in the stated source norm.

Linearized equations

To correct a residual $F(u)$ , one may first solve $DF(u)h=-F(u)$ . This cancels the displayed constant and linear terms, leaving $R(u,h)$ . Invertibility and bounds for the inverse need separate proofs. Computing $\left.\partial_\varepsilon F(u+\varepsilon h)\right|_{\varepsilon=0}$ in each direction gives a candidate linearization; directional derivatives alone do not establish Fréchet differentiability.