Convergence of a Cauchy sequence of matrices

I have a Cauchy sequence of matrices $C_i \in R^{p \times q}$, i.e. $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm (I just need the property that $\|C_1-C_2\|>\delta \Rightarrow C_1 \neq C_2$). I also know $\|C\|_{1,1} \leq t$ for a fixed $t$ (where $\|C\|_{1,1} = \sum_{i,j} |c_{ij}|$).

Can I conclude that the sequence $C_i$ also converges? (i.e., is the corresponding metric space complete?)

• What you describe (the limit of $\| C_{n+1}-C_n\|$ being $0$) does not mean that $C_n$ is Cauchy, and it doesn't imply convergence of the sequence. It is true however that the space of $p\times q$ real matrices is complete. – Olivier Bégassat Jul 30 '14 at 17:06

If a sequence in this space is Cauchy, then it converges. That is, $\Bbb R^{p \times q}$ under $\|\cdot\|_{1,1}$ (which is isometric to $\Bbb R^{pq}$ under $\|\cdot\|_1$) is indeed a complete metric space.
That being said, the condition you provided is insufficient to guarantee that the sequence is Cauchy. As a counterexample, consider the sequence in $\Bbb R^{2 \times 1}$ given by $C_n = (\cos \theta_n,\sin\theta_n)$ where $$\theta_n = \sum_{i=1}^n \frac 1i$$ Noting that $\|C_n\| < \sqrt{2}$ for each $n$.
• You could a sequence whose sequence of partial sums oscillates between $0$ and $1$, for instance, $1,-1/2,-1/2,1/4,1/4,1/4,1/4,-1/8,-1/8\dots$ – Olivier Bégassat Jul 30 '14 at 17:20
• @Milad it's equivalent to consider sequences of the form $\sum_{i=1}^n D_i$ where $D_i = C_{i+1} - C_i$. So, what you should really be looking into is the convergence of series. – Omnomnomnom Jul 30 '14 at 19:17