A question about why a space under a certain norm is complete 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)!
 A: The definition of the norm can be simplified.
$$\| Y \| = \sup \left\{|Y(t)| \exp\left(- \int_a^t |P(s)|ds\right) : a \leq t \leq b\right\}$$
Proof: Suppose $r < \| Y \|$.  Then $r < |Y(t)|\exp\left(- K\int_a^t |P(s)|ds\right)$ for some $K > 1$ and $a \leq t \leq b$.  But then $r < |Y(t)|\exp\left( - \int_a^t |P(s)|ds\right)$ and so $r$ is less than the right hand side of the equation above.  
The converse can be shown easily.
So we can rewrite the norm:
$$\| Y \| = \sup \left\{|Y(t)|f(t) : a \leq t \leq b\right\}$$
where $f : [a,b] \to [r,1]$ is a continuous decreasing surjection with $r = \exp\left(-\int_a^b |P(s)|ds\right)$.  Now suppose $\langle Y_n : n\in \mathbb{N}\rangle$ is a Cauchy sequence with respect to $\|\cdot\|$.  Then I claim this sequence has a limit, and that this limit is the entry-wise limit of the $Y_n$.  
First let's see that the entry-wise limit exists: Since $C[a,b]$ is complete, it suffices to show that the entry-wise sequences are Cauchy.  Suppose not, that is suppose for some $i,j$, we have that $(Y_n)_{i,j}$ is not Cauchy.  Then 
$$\lim _{N\to\infty}\sup\{\sup\{|(Y_m)_{i,j}(x) - (Y_n)_{i,j}(x)| : x \in [a,b]\} : m,n > N\} = L$$
for some $L > 0$.  Now fix $N > 0$.  Pick $m,n > N$ such that $\sup _{x\in [a,b]}|(Y_m)_{i,j}(x) - (Y_n)_{i,j}(x)| > L$.  So we can pick $x \in [a,b]$ such that $|(Y_m)_{i,j}(x) - (Y_n)_{i,j}(x)| > L$.  But then $|(Y_m-Y_n)(x)|>L$.  Then $|(Y_m-Y_n)(x)|f(x) > Lr$, and so $\|Y_m - Y_n\| > Lr$.  This contradicts the assumption that the $Y_n$ formed a Cauchy sequence.
Now that we know the entry-wise limit $Y$ exists, we have to show that the $Y_n$ converge to $Y$ with respect to $\|\cdot\|$.  Well
$$\|Y - Y_n\| \leq \sup _{x \in [a,b]}|Y(x) - Y_n(x)| \leq \sum_{i,j} \sup_x |Y_{i,j}(x) - (Y_n)_{i,j}(x)|$$
Since for each $i,j$ we have that $\lim _n |Y_{i,j}(x) - (Y_n)_{i,j}(x)| = 0$, the right hand side of the above inequality goes to $0$ as $n$ goes to $\infty$.
