A theorem I am reading (about the existence and uniqueness of solutions to Sturm-Liouville intial-value problems) defines a space $B$ consisting of the continuous functions defined on a closed real interval $[a,b]$ and assuming values given by complex matrices $m \times n$.
The author then defines what he calls a Bielecki norm and says that "it is clear" that with this norm $B$ is a Banach space. I am aware of the definition of a Banach space but fail to see why the claim is true. I have tried to find a limit to an arbitrary Cauchy sequence, but without luck so far.
Here it is:
$ \| Y \| = \sup \left\{|Y(t)| \exp\left(-K \int_a^t |P(s)|ds\right) : K > 1 \text{ and constant, } a \leq t \leq b\right\} $
To clarify, the Euler constant is raised to the power minus K times the integral from a to t of the norm of a fairly arbitrary complex-valued matrix with variable entries. The matrix arises from the context of the theorem, but is merely defined as being a square matrix of Lebesgue-integrable complex-valued functions. The matrix norm, in case of Y and P on the RHS of the above, is defined as the sum of absolute values of the matrix entries. This is as distinct, of course, from the Bielecki norm of Y expressed on the LHS.
Thanks if you can help (or try)!