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?)

  • $\begingroup$ 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. $\endgroup$ – 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.

In general: the finite Cartesian product of complete metric spaces will be complete (this is not true, however, for arbitrary products).

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 $n$.

  • $\begingroup$ There are also bounded counter examples, more in line with the question. $\endgroup$ – Olivier Bégassat Jul 30 '14 at 17:11
  • $\begingroup$ @OlivierBégassat I forgot about the boundedness condition. I'll rack my brain for a suitable counterexample, but if you have one in mind, perhaps you should post your own answer. $\endgroup$ – Omnomnomnom Jul 30 '14 at 17:17
  • $\begingroup$ You could a sequence whose sequence of partial sums oscillates between $0$ and $1$, for instance, $1,-1/2,-1/2,1/4,1/4,1/4,1/4,-1/8,-1/8\dots$ $\endgroup$ – Olivier Bégassat Jul 30 '14 at 17:20
  • $\begingroup$ Thanks for the answer and examples. Does this kind of sequence (where the limit of \|C_{n+1}-C_n\| is zero) have a certain name in the literature? I'd like to learn more about their properties, and perhaps sufficient conditions under which they converge. $\endgroup$ – MMKK Jul 30 '14 at 18:32
  • $\begingroup$ @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. $\endgroup$ – Omnomnomnom Jul 30 '14 at 19:17

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