Rectangular residual linearization error under a Lipschitz Jacobian

Given: The open set $U$, the $C^1$ residual $r:U\to {\mathbb{R}}^m$, the convex set $K\subseteq U$, the points $x_{\ast },x\in K$, and the Jacobian Lipschitz bound in the Statement.

Continuously differentiable maps, local inverses, and local diffeomorphisms: $C^1$ means total differentiability at every point and continuity of the derivative-matrix entries.

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: the Jacobian entries are $J_{ij}={\mathrm{\partial }}_j{r}_i$ in the standard basis.

The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$: the derivative of a composition is the composition of the derivatives.

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral: vector derivatives and integrals are componentwise; vector integration is linear.

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: a continuous scalar function on a closed bounded interval is Riemann integrable.

If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz: a vector function differentiable on a closed interval with integrable derivative has endpoint difference equal to the integral of its derivative.

For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable: for an integrable vector function, the norm of its integral is at most the integral of its norm.

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$: the Euclidean norm is induced by the standard dot product and $\mathrm{\parallel }v{\mathrm{\parallel }}_2^2=⟨v,v⟩$.

The matrix norm induced by a published vector p-norm: at $p=2$, $\mathrm{\parallel }A{\mathrm{\parallel }}_2$ is the matrix norm induced by the Euclidean vector norm.

Induced matrix norms are compatible with matrix-vector multiplication, submultiplicative, and satisfy ||I|| = 1: $\mathrm{\parallel }Av{\mathrm{\parallel }}_2\le \mathrm{\parallel }A{\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_2$ for this induced norm.

If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$: pointwise bounds between integrable real functions pass to their integrals.

Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$ and Newton–Leibniz needs only continuity on $[a,b]$, differentiability on $(a,b)$, and a Riemann-integrable extension of the interior derivative: the product rule gives $(tc)\mathrm{\prime }=c$ for a constant vector coordinate $c$, and $(t-t^2\mathrm{/}2)\mathrm{\prime }=1-t$; Newton-Leibniz gives the constant-vector integral and ${\int }_0^1(1-t)dt=1\mathrm{/}2$.