Sequence and Limit Definition at Quaternions

Question

A sequence at $\mathbb{C}$ is a map $z_n: \mathbb{N} \to \mathbb{C}$, where $z_n$ converges to a $z_0 \in \mathbb{C}$ if:

$$\forall \epsilon > 0, \text{ } \exists N \in \mathbb{N}; \text{ } \forall n \geq N \implies \left| z_n - z_0 \right| < \epsilon$$ $$\displaystyle{ \lim_{n \to \infty} z_n = z_0}$$

My question is: can we use the same definition to $\mathbb{H}$? Or $\mathbb{O}$?

$$\forall \epsilon > 0, \text{ } \exists N \in \mathbb{N}; \text{ } \forall n \geq N \implies \left| q_n - q_0 \right| < \epsilon$$ $$\displaystyle{ \lim_{n \to \infty} q_n = q_0 \in \mathbb{H}}$$

Where,

$$q_n: \mathbb{N} \to \mathbb{H}$$

Answer 1

In principle, sure. There is a natural way to associate $\mathbb{H}$ to $\mathbb{R}^4$, for $\mathbb{O}$ to correlate to $\mathbb{R}^8$, and so on, so it is very easy to let them inherit the latter's distance, in the same way $\mathbb{C}$ does from $\mathbb{R}^2$.

• To $a+bi \in \mathbb{C}$, associate $(a,b) \in \mathbb{R}^2$.
• To $a+bi+cj+dk \in \mathbb{H}$, associate $(a,b,c,d) \in \mathbb{R}^4$
• In general, send $\sum_{i=1}^{2^n} a_i \sigma_i$ to $(a_1,\cdots,a_{2^n}) \in \mathbb{R}^{2^n}$ where the $\sigma_i$ are the equivalents of $i,j,k,$ etc. in whichever number system you're working in.

Then you can just define, if $x := \sum_{i=1}^{2^n} x_i \sigma_i, y := \sum_{i=1}^{2^n} y_i \sigma_i$ lie in that number system, the metric $$ d(x,y) := \sqrt{ \sum_{i=1}^{2^n} |x_i-y_i|^2 } $$ where $|\cdot|$ is taken as in $\mathbb{R}$. This is no different than the metric for the associated $(x_i)_{i=1}^{2^n},(y_i)_{i=1}^{2^n} \in \mathbb{R}^{2^n}$, so it should work just fine.

From there, defining sequences and their convergence follows in the natural way. Let $S_n$ be that number system, the $2^n$-dimensional one over $\mathbb{R}$. Then

• A sequence $(x_n)_{n \in \mathbb{N}}$ is a map $\mathbb{N} \to S$. (This holds for pretty much any set.)

• A sequence $(x_n)_{n \in \mathbb{N}}$ (where $x_n \in S$ for each $n$) converges to $x \in S$ if this holds: $$ \forall \varepsilon > 0, \exists n \in \mathbb{N} \text{ such that } \forall n \ge N \text{ we have } d(x_n,x) < \varepsilon $$ This is more or less the same definition for any metric space $(M,d)$. I just insist on the $d(\cdot,\cdot)$ notation because I hate how $|\cdot|$ could get confused for any number of different things. If we define $\|\cdot\|_S$ to be unambiguously the distance function (or, rather, norm, but we convert it to a distance) defined previously, then we get something that might feel more "right": $$ \forall \varepsilon > 0, \exists n \in \mathbb{N} \text{ such that } \forall n \ge N \text{ we have } \|x_n-x\|_S < \varepsilon $$