On "The inverse function theorem of Nash and Moser" paper by Richard S. Hamilton he discusses the Calculus theory for continuous function $f:[a,b]\longrightarrow F$, where $F$ is a Fréchet space. In order to define the integral $\int_a^b f(t)dt$, it is used that the subspace of picewise linear maps is dense in $\mathscr{C}\left([a,b];F\right)$ (I prefer the term affine since the functions are given by $tf_0 + f_1$ for some $f_0,f_1 \in F$). To a picewise linear function $f$ that is linear over $[t_{j-1},t_j]$, he defines
$$ \int_a^b f(t) dt = \sum_{j} \frac{f(t_{j-1})+f(t_j)}{2}\left(t_j-t_{j-1}\right) $$
which is motivated by the trapezoidal area. He claims that this definition satisfies several properties that can be checked directly from the formula but I can not see how to prove the following one: given a continuous seminorm $\left\|\cdot\right\|$ in $F$,
$$ \left\|\int_a^b f(t) dt\right\| \le \int_a^b \left\|f(t)\right\| dt. $$
Suppose that $f(t)=tf_0+f_1$. Then this inequality is equivalent to
$$ \left\| \frac{a+b}{2}f_0 + f_1\right\|(b-a) \le \int_a^b \left\|tf_0+f_1\right\|dt $$
and my difficulty is dealing with the term on the right side. I've tried to use basic estimates of the norm or some "convex" estimates but didn't work. Any hint for the proof?
Take a continuous functional $g:F\to \mathbb R$ with $\|g\|_{F^*}:= \sup_{\|x\|\le1} g(x)$. Then by linearity of $g$ and construction of the integral $$ g \left( \int_a^b f(t)dt \right) = \int_a^b g(f(t))dt \le \|g\|_{F^*} \int_a^b\| f(t)\|dt. $$ Taking the sup with respect to $g$ with $\|g\|_{F^*}\le1$ yields $$ \left\| \int_a^b f(t)dt \right\| \le \int_a^b\| f(t)\|dt. $$