Proof of The Picard–Lindelöf Theorem on a Closed Interval: Existence, Uniqueness and Continuous Dependence for Globally Lipschitz Systems

We prove (i) and (ii); the matrix clauses are treated at the end. By Basic Facts about Intervals of the Real Line (v), $[a,b]$ is an interval containing the two distinct points $a$ and $b$; derivatives of maps $[a,b]\to\mathbb{R}$ are taken relative to $[a,b]$, as in Derivative of a Real Function on an Interval (i), and integrals of maps $[a,b]\to\mathbb{R}$ and $[a,b]\to\mathbb{R}^{n}$ are as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii) and Riemann Integral of a Vector-Valued or Matrix-Valued Map (ii). $(\mathbb{R}^{n},d_{E})$ is a metric space by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence (i), with $d_{E}(u,v)=\lVert u-v\rVert$ by The Euclidean Distance on $\mathbb{R}^n$ (i). Let $F$ and $L$ be as in (i).

Step 1: comparing integrals from $a$. Let $f,g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ with $f(s)\le g(s)$ for every $s\in[a,b]$, and let $t\in[a,b]$. Then $\int_{a}^{t}f$ and $\int_{a}^{t}g$ are defined, by Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (iv), and $\int_{a}^{t}f\le\int_{a}^{t}g$. Indeed, if $t=a$, both integrals are $0$ by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii). If $a<t$, then $f$ and $g$ are integrable on $[a,t]$ by A Continuous Function on a Closed Interval is Riemann Integrable (ii), that is, by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi), the restrictions $f|_{[a,t]}$ and $g|_{[a,t]}$ are bounded and integrable on $[a,t]$, with integrals $\int_{a}^{t}f$ and $\int_{a}^{t}g$; as $f|_{[a,t]}(s)\le g|_{[a,t]}(s)$ for $s\in[a,t]$, Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (iii), with $t$ in place of $b$ and these restrictions in place of $f$ and $g$, gives $\int_{a}^{t}f\le\int_{a}^{t}g$.

Step 2: the integral operator. Let $C([a,b],\mathbb{R}^{n})$ be the set of maps $[a,b]\to\mathbb{R}^{n}$ continuous on $[a,b]$, as in The Space of Continuous Maps on a Closed Interval and Its Weighted Uniform Distance (i) with $(\mathbb{R}^{n},d_{E})$ in place of $(Y,d_{Y})$. For $z\in C([a,b],\mathbb{R}^{n})$, let $h_{z}:[a,b]\to\mathbb{R}^{n}$ be the map $s\mapsto F(s,z(s))$ (which exists by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with the identity map of $[a,b]$ of Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (iii) and $z$ for its $f$ and $g$); it is continuous on $[a,b]$ by Solutions of an Initial Value Problem Are the Continuous Solutions of the Integral Equation (i), with $b$ in place of $c$. By The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps (i), with $n$ in place of $q$, $h_{z}$ in place of $x$ and $a$ in place of $c$, the integral $\int_{a}^{t}h_{z}$ is defined for every $t\in[a,b]$ and the map $t\mapsto\int_{a}^{t}h_{z}$ (which exists by Maps Given by Integrals of Vector- and Matrix-Valued Maps (i), with $[a,b]$ for its $E$ and $S$, $a$ for its $c$ and $h_{z}$ for its $x$) is differentiable; by Rules for Derivatives of Vector- and Matrix-Valued Maps (i), with $[a,b]$ in place of $I$, it is continuous at every point of $[a,b]$ relative to $[a,b]$, hence continuous on $[a,b]$ by Continuous Map Between Metric Spaces (ii). For $\zeta\in\mathbb{R}^{n}$, the map $T_{\zeta}(z):[a,b]\to\mathbb{R}^{n}$, $t\mapsto\zeta+\int_{a}^{t}h_{z}$, the sum of the constant map with value $\zeta$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i)) and that map (a map by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with the addition of $\mathbb{R}^{n}=\mathbb{R}^{[n]}$, a binary operation by Pointwise Operations on Real-Valued Maps Exist and Are Unique (i), for its $F$), is therefore continuous on $[a,b]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), so $T_{\zeta}(z)\in C([a,b],\mathbb{R}^{n})$ by The Space of Continuous Maps on a Closed Interval and Its Weighted Uniform Distance (i). This defines the map $T_{\zeta}:C([a,b],\mathbb{R}^{n})\to C([a,b],\mathbb{R}^{n})$, $z\mapsto T_{\zeta}(z)$, by Maps and Relations Given by Formulas (i) with $C([a,b],\mathbb{R}^{n})$ for its $a$ and $b$. (Here $T_{\zeta}(z)$ is the set of the pairs $(t,y)\in[a,b]\times\mathbb{R}^{n}$ for which there are $g$ and $r$ with $g=h_{z}$, $r=\int_{a}^{t}g$ and $y=\zeta+r$; on $z$, $t$, $y$, $g$ and $r$, with $F$, $\zeta$ and the addition of $\mathbb{R}^{n}$ as further parameters, the condition $g=h_{z}$ is expressed by the predicative formula $\mu(g,[a,b],\mathbb{R}^{n})\wedge\forall s\,\forall v\,\forall r'\,\bigl(((s,v)\in z\wedge((s,v),r')\in F)\Rightarrow(s,r')\in g\bigr)$, with $\mu$ as in Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (vii), by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (i) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv); the condition that $\int_{a}^{t}g$ is defined and equals $r$ by the formula of Maps Given by Integrals of Vector- and Matrix-Valued Maps (iii), with $[a,b]$ for its $E$, $n$ for its $q$ and $g$, $a$, $t$ and $r$ for its $x$, $u$, $v$ and $y$ (the variables renamed as in Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (i)); and $y=\zeta+r$ by Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (iv); so the set $T_{\zeta}(z)$ is given by Sets and Maps: Ordinary Notation (iii) from a predicative formula in $z$, by Predicativity of Expressions with Abbreviations, and Symbol-Free Forms of the Basic Defined Set Symbols (i), the guard on $z$ being $z\in C([a,b],\mathbb{R}^{n})$.) By Solutions of an Initial Value Problem Are the Continuous Solutions of the Integral Equation (ii), with $b$ in place of $c$, a map $z:[a,b]\to\mathbb{R}^{n}$ is a solution of the initial value problem with right-hand side $F$ and initial value $\zeta$ if and only if $z\in C([a,b],\mathbb{R}^{n})$ and $z(t)=T_{\zeta}(z)(t)$ for every $t\in[a,b]$, that is, $T_{\zeta}(z)=z$.

Step 3: the difference estimate. Let $z_{1},z_{2}\in C([a,b],\mathbb{R}^{n})$, let $\zeta_{1},\zeta_{2}\in\mathbb{R}^{n}$, and let $m:[a,b]\to\mathbb{R}$ be the map $s\mapsto\lVert z_{1}(s)-z_{2}(s)\rVert$ (which exists by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with $z_{1}$ and $z_{2}$ for its $f$ and $g$ and the map $(u,v)\mapsto\lVert u-v\rVert$ of The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps (iv) for its $F$). We show that $m$ is continuous on $[a,b]$ and that, for every $t\in[a,b]$, the integral $\int_{a}^{t}m$ is defined and

$$\bigl\lVert T_{\zeta_{1}}(z_{1})(t)-T_{\zeta_{2}}(z_{2})(t)-(\zeta_{1}-\zeta_{2})\bigr\rVert\le L\int_{a}^{t}m .$$

The constant map $[a,b]\to\mathbb{R}$ with value $-1$ is continuous at every point of $[a,b]$ relative to $[a,b]$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space (i), with $(\mathbb{R},d_{\mathbb{R}})$ and $[a,b]$ in place of $(X,d)$ and $A$, hence continuous on $[a,b]$ by Continuous Map Between Metric Spaces (ii). Therefore the maps $s\mapsto z_{1}(s)-z_{2}(s)=z_{1}(s)+(-1)z_{2}(s)$ and $g:[a,b]\to\mathbb{R}^{n}$, $s\mapsto h_{z_{1}}(s)-h_{z_{2}}(s)=h_{z_{1}}(s)+(-1)h_{z_{2}}(s)$, are continuous on $[a,b]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), and so are $m$ and $s\mapsto\lVert g(s)\rVert$ (the composite of $g$ with the map $u\mapsto\lVert u\rVert$ of The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps (iii), by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)), by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iii) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v). Let $t\in[a,b]$. By Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (iii) and Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (ii), with $[a,b]$ in place of $I$ and $a$ and $t$ in place of $u$ and $v$, $\int_{a}^{t}g$ is defined and equals $\int_{a}^{t}h_{z_{1}}-\int_{a}^{t}h_{z_{2}}$, which is $T_{\zeta_{1}}(z_{1})(t)-T_{\zeta_{2}}(z_{2})(t)-(\zeta_{1}-\zeta_{2})$. By Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (viii), with $a\le t$, the integral $\int_{a}^{t}\lVert g(s)\rVert\,ds$ is defined and $\bigl\lVert\int_{a}^{t}g\bigr\rVert\le\int_{a}^{t}\lVert g(s)\rVert\,ds$. By the Lipschitz condition of (i), $\lVert g(s)\rVert\le L\,m(s)$ for every $s\in[a,b]$, and the map $Lm$ is continuous on $[a,b]$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space (iv) and Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v); so Step 1 gives $\int_{a}^{t}\lVert g(s)\rVert\,ds\le\int_{a}^{t}Lm$. Finally $\int_{a}^{t}m$ is defined by Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (iv), and $\int_{a}^{t}Lm=L\int_{a}^{t}m$ by Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (ii). Combining these gives the estimate.

Step 4: the weight. Let $\theta:[a,b]\to\mathbb{R}$ be the map $t\mapsto2L(t-a)=(2L)t+(-2La)$ (the map $(2L)\,\mathrm{id}_{[a,b]}+k$, with the identity map of Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (iii), read as a map into $\mathbb{R}$, as its range lies in $[a,b]\subseteq\mathbb{R}$ (Functions, Values of a Function, and Functions from One Class to Another (iii)), the constant map $k$ with value $-2La$ of Pointwise Operations on Real-Valued Maps (ii), and the scalar multiple and sum of Pointwise Operations on Real-Valued Maps (i)). By Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (ii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i) and Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iv), $\theta$ is differentiable at every $t\in[a,b]$ with $\theta'(t)=2L$. By Basic Facts about Intervals of the Real Line (i), $\mathbb{R}$ is an interval containing two distinct points, and by Basic Properties of the Exponential Function (vii), with $\mathbb{R}$ in place of $I$, $\exp$ is differentiable at every $r\in\mathbb{R}$ with derivative $\exp(r)$ (that clause is about $\exp|_{\mathbb{R}}$, which is $\exp$ by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iii)). By Chain Rule for Real Functions on Intervals (i), with $[a,b]$ and $\mathbb{R}$ in place of $I$ and $J$, $\theta$ in place of $f$ and $\exp$ in place of $g$, the map $\varphi=\exp\circ\theta:[a,b]\to\mathbb{R}$, $\varphi(t)=\exp\bigl(2L(t-a)\bigr)$, is differentiable at every $t\in[a,b]$ with $\varphi'(t)=2L\,\varphi(t)$. By The Increment Form of the Derivative, and Differentiability Implies Continuity (ii) and Continuous Map Between Metric Spaces (ii), $\varphi$ is continuous on $[a,b]$; $\varphi(t)>0$ for every $t\in[a,b]$ by Basic Properties of the Exponential Function (iii), and $\varphi(a)=\exp(0)=1$ by Basic Properties of the Exponential Function (i). Let $w=1/\varphi:[a,b]\to\mathbb{R}$; it is continuous on $[a,b]$ by Continuity of Identity Maps, Compositions and Reciprocals (iii), $w(t)>0$ and $w(t)\varphi(t)=1$ for every $t\in[a,b]$.

For $\delta\in\mathbb{R}$, let $\psi_{\delta}=\tfrac{\delta}{2}\varphi:[a,b]\to\mathbb{R}$, the scalar multiple of Pointwise Operations on Real-Valued Maps (i). By Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v), $\psi_{\delta}$ is differentiable at every $t\in[a,b]$, hence differentiable on $[a,b]$, with $\psi_{\delta}'(t)=L\delta\,\varphi(t)$; its derivative $\psi_{\delta}'$ (Derivative of a Real Function on an Interval (iv)) is the map $L\delta\,\varphi$, continuous on $[a,b]$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space (iv) and Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v). Hence, for $t\in[a,b]$, $\int_{a}^{t}\psi_{\delta}'$ is defined by Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (iv), and by Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (vi)

$$\int_{a}^{t}\psi_{\delta}'=\psi_{\delta}(t)-\psi_{\delta}(a)=\tfrac{\delta}{2}\bigl(\varphi(t)-1\bigr).$$

Step 5: proof of (i). Let $\xi\in\mathbb{R}^{n}$, and let $\mathcal{C}=C([a,b],\mathbb{R}^{n})$ carry the weighted uniform distance $d_{w}$ of The Space of Continuous Maps on a Closed Interval and Its Weighted Uniform Distance (ii), with $(\mathbb{R}^{n},d_{E})$ in place of $(Y,d_{Y})$ and the weight $w$ of Step 4. By The Weighted Uniform Distance Is a Metric, and Is Complete When the Target Is Complete (i), $(\mathcal{C},d_{w})$ is a metric space; it is nonempty by The Weighted Uniform Distance Is a Metric, and Is Complete When the Target Is Complete (ii), as $0\in\mathbb{R}^{n}$; and it is complete by The Weighted Uniform Distance Is a Metric, and Is Complete When the Target Is Complete (iii), as $(\mathbb{R}^{n},d_{E})$ is complete by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence (iv). Let $T=T_{\xi}:\mathcal{C}\to\mathcal{C}$ be the map of Step 2.

$T$ is a contraction with constant $\tfrac12$. Let $x,y\in\mathcal{C}$ and $\delta=d_{w}(x,y)$, and let $m:[a,b]\to\mathbb{R}$, $s\mapsto\lVert x(s)-y(s)\rVert$, be the map of Step 3 for $z_{1}=x$ and $z_{2}=y$. By The Space of Continuous Maps on a Closed Interval and Its Weighted Uniform Distance (iii), $w(s)\,m(s)\le\delta$ for every $s\in[a,b]$; multiplying by $\varphi(s)>0$ and using $w(s)\varphi(s)=1$ gives $m(s)\le\delta\,\varphi(s)$, hence, as $L\ge0$, $L\,m(s)\le L\delta\,\varphi(s)=\psi_{\delta}'(s)$. Moreover $\delta\ge w(a)\,m(a)\ge0$, by The Space of Continuous Maps on a Closed Interval and Its Weighted Uniform Distance (iii) and Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i). Let $t\in[a,b]$. By Step 3 with $\zeta_{1}=\zeta_{2}=\xi$, by Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (ii), by Step 1 applied to $Lm$ and $\psi_{\delta}'$, and by Step 4,

$$\lVert T(x)(t)-T(y)(t)\rVert\le L\int_{a}^{t}m=\int_{a}^{t}Lm\le\int_{a}^{t}\psi_{\delta}'=\tfrac{\delta}{2}\bigl(\varphi(t)-1\bigr)\le\tfrac{\delta}{2}\,\varphi(t),$$

the last step because $\delta\ge0$. Multiplying by $w(t)>0$ gives $w(t)\,d_{E}\bigl(T(x)(t),T(y)(t)\bigr)\le\tfrac{\delta}{2}$. By The Space of Continuous Maps on a Closed Interval and Its Weighted Uniform Distance (iii), applied to $T(x)$ and $T(y)$, there is $t_{0}\in[a,b]$ with $d_{w}(T(x),T(y))=w(t_{0})\,d_{E}\bigl(T(x)(t_{0}),T(y)(t_{0})\bigr)$, so $d_{w}(T(x),T(y))\le\tfrac12\,d_{w}(x,y)$. Since $0\le\tfrac12<1$, $T$ is a contraction by Contraction of a Metric Space (i).

Existence and uniqueness. By Contraction Mapping Theorem on a Nonempty Complete Metric Space (i), applied to $(\mathcal{C},d_{w})$ and $T$, there is exactly one $x\in\mathcal{C}$ with $T(x)=x$. By Step 2, with $\xi$ in place of $\zeta$, $x$ is a solution of the initial value problem with right-hand side $F$ and initial value $\xi$; and every solution $z$ of it lies in $\mathcal{C}$ and satisfies $T(z)=z$, so $z=x$. This proves (i).

Step 6: proof of (ii). Let $\xi,\eta\in\mathbb{R}^{n}$, and let $x$ and $y$ be the solutions with initial values $\xi$ and $\eta$, which exist and are unique by (i). By Step 2, $x,y\in C([a,b],\mathbb{R}^{n})$, $T_{\xi}(x)=x$ and $T_{\eta}(y)=y$. Let $m:[a,b]\to\mathbb{R}$, $s\mapsto\lVert x(s)-y(s)\rVert$, be the map of Step 3 for $z_{1}=x$, $z_{2}=y$, $\zeta_{1}=\xi$ and $\zeta_{2}=\eta$; it is continuous on $[a,b]$. For $t\in[a,b]$, writing $x(t)-y(t)=(\xi-\eta)+\bigl(x(t)-y(t)-(\xi-\eta)\bigr)$, Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (vi) and Step 3 give

$$m(t)\le\lVert\xi-\eta\rVert+\bigl\lVert T_{\xi}(x)(t)-T_{\eta}(y)(t)-(\xi-\eta)\bigr\rVert\le\lVert\xi-\eta\rVert+L\int_{a}^{t}m .$$

By Gronwall's Lemma (Integral Form) (i), with $m$ in place of $u$, $\alpha=\lVert\xi-\eta\rVert$ and $\beta=L\ge0$, $m(t)\le\lVert\xi-\eta\rVert\exp\bigl(L(t-a)\bigr)$ for every $t\in[a,b]$, which is (ii).

Proof of (iii) and (iv). Steps 1 and 4 do not involve $F$. Steps 2, 3, 5 and 6 apply word for word to the $F$ and $L$ of (iii), with $\mathcal{M}_{p\times q}(\mathbb{R})$, $d_{p\times q}$, the operator norm, $\Xi$ and $\Theta$, the solutions $X$ and $Y$, the zero matrix $0_{p\times q}$, The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii) and (iii) in place of $\mathbb{R}^{n}$, $d_{E}$, the Euclidean norm, $\xi$ and $\eta$, $x$ and $y$, $0$, The Initial Value Problem for a First-Order System of Ordinary Differential Equations (i) and (i); using that $(\mathcal{M}_{p\times q}(\mathbb{R}),d_{p\times q})$ is a metric space by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence (i) and complete by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence (vii), with $d_{p\times q}(A,B)=\lVert A-B\rVert$ by The Distance between Real Matrices (i); that $\lVert A\rVert\ge0$ and $\lVert A+B\rVert\le\lVert A\rVert+\lVert B\rVert$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (i) and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (vi), in place of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i) and Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (vi); and the matrix halves of Solutions of an Initial Value Problem Are the Continuous Solutions of the Integral Equation (i), The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps (i), Rules for Derivatives of Vector- and Matrix-Valued Maps (i), Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv), Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iii), Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (iii), Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (ii) and Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (viii), together with Solutions of an Initial Value Problem Are the Continuous Solutions of the Integral Equation (iii) in place of Solutions of an Initial Value Problem Are the Continuous Solutions of the Integral Equation (ii), Maps Given by Integrals of Vector- and Matrix-Valued Maps (ii) in place of Maps Given by Integrals of Vector- and Matrix-Valued Maps (i), the matrix half of Maps Given by Integrals of Vector- and Matrix-Valued Maps (iii) (with $\mu(g,[a,b],\mathcal{M}_{p\times q}(\mathbb{R}))$ in the formula for $g=h_{z}$), the operator norm as a map $\mathcal{M}_{p\times q}(\mathbb{R})\to\mathbb{R}$ by The Operator Norm of a Real Matrix (i) and the map $d_{p\times q}$, $(A,B)\mapsto\lVert A-B\rVert$, of The Distance between Real Matrices (i) in place of the maps of The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps (iii) and The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps (iv), the addition of $\mathcal{M}_{p\times q}(\mathbb{R})=\mathbb{R}^{[p]\times[q]}$ in place of that of $\mathbb{R}^{[n]}$, and Riemann Integral of a Vector-Valued or Matrix-Valued Map (iv) in place of Riemann Integral of a Vector-Valued or Matrix-Valued Map (ii).