Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals

Let $p,q,r\ge1$ be natural numbers. For $x$ in the Euclidean space $\mathbb{R}^{p}$ write $|x|=d(x,0)$ with the Euclidean distance $d$, so that $d(x,y)=|x-y|$; for a real $p\times q$ matrix $X$ write $|X|$ for the Euclidean norm of the tuple of its entries and $|X|_{e}=\max_{i,j}|X_{ij}|$. Products are the matrix product and matrix-vector product, $(\cdot)^{\top}$ is the transpose, and $\cdot$ is the dot product.

1. (Entry and norm inequalities) For $x\in\mathbb{R}^{p}$ and each $i$: $|x^{i}|\le|x|\le\sum_{l=1}^{p}|x^{l}|\le p\max_{l}|x^{l}|$. For a real $p\times q$ matrix $X$ and all $i,j$: $|X_{ij}|\le|X|_{e}$, $|X_{ij}|\le|X|$, and $|X|\le pq\,|X|_{e}$.

2. (Product entry bound) For a real $p\times q$ matrix $U$ and a real $q\times r$ matrix $V$: $|(UV)_{il}|\le q\,|U|_{e}\,|V|_{e}$ for all $i,l$; in particular $|UV|_{e}\le q\,|U|_{e}\,|V|_{e}$.

3. (Transpose identities) For matrices $U$ ($p\times q$) and $V$ ($q\times r$): $(UV)^{\top}=V^{\top}U^{\top}$. For a real $p\times q$ matrix $M$, $y\in\mathbb{R}^{p}$, and $z\in\mathbb{R}^{q}$: $y\cdot(Mz)=(M^{\top}y)\cdot z$.

4. (Indefinite Riemann integrals) Let $a<b$ be real numbers and $\varphi:[a,b]\to\mathbb{R}$ continuous on $[a,b]$, the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric, with the degenerate-interval convention of Mean-Square Riemann Integral of a Family of Random Variables. Then for all $a\le s\le t\le b$, with the Riemann integral (existing by claim 3 of the integral toolkit on a compact interval),

the maximum existing by Extreme Value Theorem on a Compact Subset of a Metric Space, the interval $[a,b]$ being nonempty, since $a<b$, and a compact subset of the real line by Closed Interval $[a,b]$ is Compact in $\mathbb{R}$; consequently $t\mapsto\int_a^t\varphi(u)\,du$ is continuous on all of $[a,b]$, including the endpoints.

5. (Vector integral bound) For $w:[a,b]\to\mathbb{R}^{p}$ with continuous components and $a\le t\le b$: $|w(\cdot)|$ and the $|w^{i}(\cdot)|$ are continuous, and

the integral of $w$ being taken componentwise.

6. (Bilinear forms under entrywise integrals) For an assignment $U$ of a real $p\times q$ matrix $U(t)$ with entries continuous on $[a,b]$, $y\in\mathbb{R}^{p}$, $z\in\mathbb{R}^{q}$, and $a\le t\le b$:

the integral of $U$ being taken entrywise.

7. (Translation) For $u:[a,b]\to\mathbb{R}$ continuous and $t\in[a,b]$: the function $\tau\mapsto u(a+\tau)$ is continuous on $[0,b-a]$ and