Proof of Rules for the Riemann Integral of Vector- and Matrix-Valued Maps

Throughout, for $u,v\in I$ with $u\le v$ we have $[u,v]\subseteq I$ by Basic Facts about Intervals of the Real Line (ii). The component maps $x_{k}:I\to\mathbb{R}$ ($k\in[q]$) and the entry maps $A_{ij}:I\to\mathbb{R}$ are those of Riemann Integral of a Vector-Valued or Matrix-Valued Map (i) and Riemann Integral of a Vector-Valued or Matrix-Valued Map (iii). Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations is applied with $(X,d)=(\mathbb{R},d_{\mathbb{R}})$ and $S=I$.

Step 1 (real-valued maps). For maps $I\to\mathbb{R}$ we use Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative with $E=I$: Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (i) and Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (ii) for sums and scalar multiples, Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (iii) for constant maps (its hypothesis holds, as $[u,v]\subseteq I$ if $u<v$ and $[v,u]\subseteq I$ if $v<u$), and, for maps continuous on $I$, Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (iv) and Rules for Oriented Riemann Integrals of Real-Valued Maps: Sums, Scalar Multiples, Constants, Continuous Maps, Additivity and Evaluation by an Antiderivative (v). For $u<v$ in $I$, that $\int_{u}^{v}f$ is defined means that $f$ is integrable on $[u,v]$ (The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii)).

Step 2 (finite linear combinations of real maps). Let $u,v\in I$ and $r\in\mathbb{N}$, let $j\mapsto c_{j}$ be a map from $[r]$ to $\mathbb{R}$, and let $(f_{j})_{j\in[r]}$ be a family of maps $I\to\mathbb{R}$ such that every $\int_{u}^{v}f_{j}$ is defined. The set $[r]$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vii) and contains $1$ by The Segments [n]: Each Is a Set, and 1 Lies in [n] for n ∈ ℕ (ii); so, by Finite Sums of Real-Valued Maps: Continuity, Integral and Derivative (ii) with $J=[r]$ and $E=I$, the integral from $u$ to $v$ of the map $I\to\mathbb{R}$, $t\mapsto\sum_{j=1}^{r}c_{j}f_{j}(t)$, is defined and equals $\sum_{j=1}^{r}c_{j}\int_{u}^{v}f_{j}$. Below, the families are the component family $k\mapsto x_{k}$ of Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (i) and rows $j\mapsto A_{ij}$ and columns $j\mapsto A_{jk}$ of the entry family $(i,j)\mapsto A_{ij}$ of Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (ii), and the coefficients are the components of $z$ or a row or column of a fixed matrix, a map on $[m]\times[q]$, $[m]\times[p]$ or $[q]\times[n]$ (Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i)); a row $j\mapsto G(i,j)$ or column $j\mapsto G(j,k)$ of a map $G$ on a product is the composite of $G$ with $j\mapsto(i,j)$, respectively $j\mapsto(j,k)$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (ii), applied to an identity map, Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (iii), and a constant map, Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i); then Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)).

Step 3 (components). For $t\in I$, $x(t)\in\mathbb{R}^{q}$ and $A(t)\in\mathcal{M}_{p\times q}(\mathbb{R})$ by Values of a Map between Classes, and Distances in a Metric Space (iii), so their components and entries are defined. By Riemann Integral of a Vector-Valued or Matrix-Valued Map (ii), for $u,v\in I$ the integral $\int_{u}^{v}x$ is defined if and only if every $\int_{u}^{v}x_{k}$ ($k\in[q]$) is defined, and then $\bigl(\int_{u}^{v}x\bigr)_{k}=\int_{u}^{v}x_{k}$; by Riemann Integral of a Vector-Valued or Matrix-Valued Map (iv) the same holds for matrix-valued maps with entries in place of components. Two vectors, respectively two matrices of the same size, are equal when their components, respectively entries, agree (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iii), Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iii)). In each clause below the vector and the matrix statement are proved by the same argument, applied to the components $x_{k}$, respectively to the entries $A_{ij}$; we write out the vector case and indicate the changes.

Proof of (i). Let $x$ be continuous on $I$. By Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), in the form of Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (i) with $S=I$, every $x_{k}$ is continuous on $I$, so $\int_{u}^{v}x_{k}$ is defined for all $u,v\in I$, and $x_{k}$ is integrable on $[u,v]$ for $u<v$ in $I$ (Step 1). By Riemann Integral of a Vector-Valued or Matrix-Valued Map (i) and Step 3 the clause follows for $x$. For $A$, use Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (ii) and Riemann Integral of a Vector-Valued or Matrix-Valued Map (iii).

Proof of (ii) and (iii). By Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv), the $k$-th component maps of $t\mapsto x(t)+y(t)$ and $t\mapsto c\,x(t)$ are $x_{k}+y_{k}$ and $c\,x_{k}$. If $\int_{u}^{v}x$ and $\int_{u}^{v}y$ are defined, then so are $\int_{u}^{v}x_{k}$ and $\int_{u}^{v}y_{k}$ (Step 3), and Step 1 gives that $\int_{u}^{v}(x_{k}+y_{k})$ and $\int_{u}^{v}c\,x_{k}$ are defined, with values $\int_{u}^{v}x_{k}+\int_{u}^{v}y_{k}$ and $c\int_{u}^{v}x_{k}$, the $k$-th components of $\int_{u}^{v}x+\int_{u}^{v}y$ and $c\int_{u}^{v}x$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv)). Step 3 gives both clauses for $x$, $y$. For $A$, $B$ use Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv).

Proof of (iv). Let $x$ be continuous on $I$ and $u,v,w\in I$. Each $x_{k}$ is continuous on $I$, as shown in the proof of (i), so by Step 1 the integrals $\int_{u}^{w}x_{k}$, $\int_{u}^{v}x_{k}$ and $\int_{v}^{w}x_{k}$ are defined and $\int_{u}^{w}x_{k}=\int_{u}^{v}x_{k}+\int_{v}^{w}x_{k}$. By Step 3 and Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv) this gives the clause for $x$; for $A$ use the entries and Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv).

Proof of (v). The $k$-th component map of $t\mapsto b$ is the constant map with value $b_{k}$. By Step 1, $\int_{u}^{v}b_{k}\,dt$ is defined and equals $(v-u)\,b_{k}$, the $k$-th component of $(v-u)\,b$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv)); Step 3 gives the claim. For $C$ use the entries and Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv).

Proof of (vi). Let $\int_{u}^{v}x$ be defined. For $i\in[m]$ and $t\in I$, $(Lx(t))_{i}=\sum_{j=1}^{q}L_{ij}x_{j}(t)$ by Matrix-Vector Product (i). By Step 2 with $r=q$, $c_{j}=L_{ij}$ and $f_{j}=x_{j}$, the integral from $u$ to $v$ of this $i$-th component map is defined and equals $\sum_{j=1}^{q}L_{ij}\int_{u}^{v}x_{j}=\sum_{j=1}^{q}L_{ij}\bigl(\int_{u}^{v}x\bigr)_{j}=\bigl(L\int_{u}^{v}x\bigr)_{i}$, by Step 3 and Matrix-Vector Product (i); Step 3 gives the first identity. Likewise $z\cdot x(t)=\sum_{k=1}^{q}z_{k}x_{k}(t)$ by The Dot Product on $\mathbb{R}^n$ (i), and Step 2 gives that $\int_{u}^{v}z\cdot x$ is defined and equals $\sum_{k=1}^{q}z_{k}\int_{u}^{v}x_{k}=z\cdot\int_{u}^{v}x$.

Let $\int_{u}^{v}A$ be defined, and write $Y=\int_{u}^{v}A$, so $Y_{ij}=\int_{u}^{v}A_{ij}$ by Step 3. By Matrix-Vector Product (i) and Product of Real Matrices (i), for $t\in I$,

$$(A(t)z)_{i}=\sum_{j=1}^{q}z_{j}A_{ij}(t),\qquad(MA(t))_{ik}=\sum_{j=1}^{p}M_{ij}A_{jk}(t),\qquad(A(t)N)_{il}=\sum_{j=1}^{q}N_{jl}A_{ij}(t),$$

and Step 2 shows that the integrals of these entry maps from $u$ to $v$ are defined and equal $\sum_{j}z_{j}Y_{ij}=(Yz)_{i}$, $\sum_{j}M_{ij}Y_{jk}=(MY)_{ik}$ and $\sum_{j}Y_{ij}N_{jl}=(YN)_{il}$, by the same definitions. Step 3 gives the first three identities. Finally, the map $t\mapsto MA(t)$, from $I$ to $\mathcal{M}_{m\times q}(\mathbb{R})$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i) and Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with the matrix product as the map of The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (i)), has a defined integral from $u$ to $v$, equal to $MY$, as just shown. The identity for $A(t)N$, re-instantiated with $m$ in place of $p$ and with $t\mapsto MA(t)$ in place of $A$, shows that $\int_{u}^{v}\bigl(MA(t)\bigr)N\,dt$ is defined and equals $(MY)N$.

Proof of (vii). By Transpose of a Real Matrix (i), $(A(t)^{\top})_{ji}=A_{ij}(t)$ for $t\in I$, $i\in[p]$, $j\in[q]$, so the $(j,i)$ entry map of $t\mapsto A(t)^{\top}$ is $A_{ij}$. If $\int_{u}^{v}A$ is defined, Step 3 shows that $\int_{u}^{v}A(t)^{\top}dt$ is defined with $(j,i)$ entry $\int_{u}^{v}A_{ij}=\bigl(\int_{u}^{v}A\bigr)_{ij}=\bigl((\int_{u}^{v}A)^{\top}\bigr)_{ji}$, which is the claim.

Proof of (viii), vectors. Let $x$ be continuous on $I$ and $u\le v$ in $I$. By Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), in the form of Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iii) with $S=I$, the map $\lVert x\rVert:t\mapsto\lVert x(t)\rVert$ is continuous on $I$, so $\int_{u}^{v}\lVert x(t)\rVert\,dt$ is defined by Step 1; $\int_{u}^{v}x$ is defined by (i). If $u=v$, both sides are $0$: $\int_{u}^{u}x=0$, the origin, by Step 3, The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii) and Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (v), $\lVert0\rVert=0$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iii), and $\int_{u}^{u}\lVert x(t)\rVert\,dt=0$. Let $u<v$ and $y_{0}=\int_{u}^{v}x$. By (vi), $\int_{u}^{v}y_{0}\cdot x$ is defined and

$$\int_{u}^{v}y_{0}\cdot x(t)\,dt=y_{0}\cdot y_{0}=\lVert y_{0}\rVert^{2}$$

by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i). For $t\in[u,v]$, $y_{0}\cdot x(t)\le|y_{0}\cdot x(t)|\le\lVert y_{0}\rVert\,\lVert x(t)\rVert$ by Rules of Arithmetic and Order in an Ordered Field (x) and The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ (i). The restrictions to $[u,v]$ of $y_{0}\cdot x$ and of $\lVert x\rVert$ are bounded and integrable on $[u,v]$ (The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi)), and so is $\lVert y_{0}\rVert$ times the latter, with integral $\lVert y_{0}\rVert\int_{u}^{v}\lVert x(t)\rVert\,dt$, by Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (ii). Hence Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (iii), applied on $[u,v]$, gives

$$\lVert y_{0}\rVert^{2}\le\lVert y_{0}\rVert\int_{u}^{v}\lVert x(t)\rVert\,dt .$$

The constant map $[u,v]\to\mathbb{R}$ with value $0$ is bounded and integrable with integral $0\cdot(v-u)=0$ by Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (vi), and $0\le\lVert x(t)\rVert$ for $t\in[u,v]$ (Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i)); so Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (iii), applied on $[u,v]$ to this constant map and the restriction of $\lVert x\rVert$, gives $0\le\int_{u}^{v}\lVert x(t)\rVert\,dt$. If $\lVert y_{0}\rVert=0$ the claim follows from this; otherwise $\lVert y_{0}\rVert>0$, so $\lVert y_{0}\rVert^{-1}>0$, and multiplying by $\lVert y_{0}\rVert^{-1}$ gives $\lVert y_{0}\rVert\le\int_{u}^{v}\lVert x(t)\rVert\,dt$ (Rules of Arithmetic and Order in an Ordered Field (viii), Rules of Arithmetic and Order in an Ordered Field (vi)).

Proof of (viii), matrices. Let $A$ be continuous on $I$ and $u\le v$ in $I$. By Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), in the form of Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iii) with $S=I$, the map $t\mapsto\lVert A(t)\rVert$ is continuous on $I$, so $\int_{u}^{v}\lVert A(t)\rVert\,dt$ is defined by Step 1, and $\int_{u}^{v}A$ is defined by (i). If $u=v$, then $\int_{u}^{u}A=0_{p\times q}$ by Step 3, The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii) and Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (v), whose operator norm is $0$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (iv), and $\int_{u}^{u}\lVert A(t)\rVert\,dt=0$. Let $u<v$, $Y=\int_{u}^{v}A$ and $\kappa=\int_{u}^{v}\lVert A(t)\rVert\,dt$, and let $z\in\mathbb{R}^{q}$. By Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), in the form of Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) with $S=I$, the constant map $I\to\mathbb{R}^{q}$, $t\mapsto z$, is continuous on $I$, and hence so is $t\mapsto A(t)z$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i) and Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with the map of The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (ii)). By (vi), $\int_{u}^{v}A(t)z\,dt=Yz$, and by the vector case, with $p$ in place of $q$, applied to $t\mapsto A(t)z$,

$$\lVert Yz\rVert\le\int_{u}^{v}\lVert A(t)z\rVert\,dt .$$

For $t\in[u,v]$, $\lVert A(t)z\rVert\le\lVert A(t)\rVert\,\lVert z\rVert$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (ii). As $u<v$ and the integrals $\int_{u}^{v}\lVert A(t)z\rVert\,dt$ (by the vector case) and $\kappa$ are defined, the restrictions to $[u,v]$ of $t\mapsto\lVert A(t)z\rVert$ and of $t\mapsto\lVert A(t)\rVert$ are bounded and integrable on $[u,v]$, by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi); the restriction to $[u,v]$ of $t\mapsto\lVert z\rVert\,\lVert A(t)\rVert$ is bounded and integrable with integral $\lVert z\rVert\,\kappa$ by Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (ii), so Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants (iii) gives $\int_{u}^{v}\lVert A(t)z\rVert\,dt\le\kappa\,\lVert z\rVert$. Thus $\lVert Yz\rVert\le\kappa\,\lVert z\rVert$ for every $z\in\mathbb{R}^{q}$, and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (iii) gives $\lVert Y\rVert\le\kappa$.