Let $f : [a, b] \to \mathbb R^{m \times n}$ be a continuous matrix valued function on the compact interval $[a, b]$. Let $X = C([a, b], \mathbb R^n)$ and $Y = C([a, b], \mathbb R^m)$ be spaces of continuous functions.
Let $F: X \to Y$ such that $(F \phi)(t) = f(t) * \phi(t)$. I want to show that $||A||_{op} = \sup_{t \in [a, b]} {|f(t)|}$, where the norm of the matrix is just the operator norm.
It is pretty easy to show that $||F||_{op} \leq \sup_{t \in [a, b]} {|f(t)|}$, but I don't know how to show that it is in fact an equality.
Thanks!