Inequality involving norm of matrix integral This question seems basic but I could not find an answer. I have seen the inequality
$$\left\|\int_a^b x(t) dt \right\| \leq \int_a^b \left\| x(t) \right\| dt $$
where $x(t) \in \mathbb{R}^n$ is a vector function and $\|\cdot\|$ is a vector norm, and $a < b$.
I wonder if this also holds for matrices with induced norm, that is
$$\left\|\int_a^b X(t) dt \right\| \leq \int_a^b \left\| X(t) \right\| dt $$
where $X(t)$ is a matrix function and $\|\cdot\|$ is an induced matrix norm, and $a < b$.  If it is true, is there any reliable citation source?
 A: If Riemann-integrals are good enough for you, then these inequalities are just the disguised triangle inequality (here with sloppy notation):
$$
\left\|\int_a^b X(t) dt \right\| 

= \left\| \lim_{\mathcal Z} \sum_{\mathcal Z}X(\xi)  \right\|

\leq  \lim_{\mathcal Z} \sum_{\mathcal Z} \left\| X(\xi)  \right\|


\leq \int_a^b \left\| X(t) \right\| dt$$
A: If $||\cdot||$ is given by $||A||=\sup_{v\neq 0}\frac{N(Av)}{N(v)}$ then we have for a fixed $v$
$$N\left(\int_a^bX(t)dtv\right)=N\left(\int_a^bX(t)vdt\right)\leq \int_a^bN\left(X(t)v\right)dt\leq \int_a^b||X(t)||N(v)dt$$
so $\|\int_a^b X(t)dt\|\leq \int_a^b||X(t)||dt$. 
In fact more generally, if $f\colon [a,b]\to X$ where $(X,||\dot||$ is a Banach space then $\|\int_a^b f(t)dt\|\leq \int_a^b||f(t)||dt$. It can be showed using a corollary of Hahn-Banach theorem:
\begin{align*}\|\int_a^b f(t)dt\|&=\sup_{\varphi\in X',||\varphi||=1}\left|\varphi\left(\int_a^b f(t)dt\right)\right|\\
&=\sup_{\varphi\in X',||\varphi||=1}\left|\int_a^b \varphi\left(f(t)\right)dt\right|\\
&\leq \sup_{\varphi\in X',||\varphi||=1}\int_a^b \left|\varphi\left(f(t)\right)\right|dt\\
&\leq 
\sup_{\varphi\in X',||\varphi||=1}\int_a^b \|f(t)\|dt
\\
&=\int_a^b \|f(t)\|dt.
\end{align*}
