# Total Internal Reflection

Technology and Art

# A Quick Note on Proving the Triangle Inequality on a Derived Distance Metric using Monotonicity

Avishek Sen Gupta on 20 October 2022

This is a quick note on proving the Triangle Inequality criterion of the following claim:

If $$d(x,y)$$ is a distance metric, then $$\bar{d}(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ is also a valid distance metric.

The four criteria for satisfying a distance metric are:

• (M1) $$0 \leq d(x,y)<\infty, d(x,y)\in \mathbb{R}$$
• (M2) $$d(x,y)=0$$ if and only if $$x=y$$
• (M3) $$d(x,y)=d(y,x)$$
• (M4) $$d(x,z) \leq d(x,y) + d(y,z)$$

(M1) to (M3) follow quite readily. Let us look at proving (M4).

Observing the form of $$\bar{d}(x,y)$$, let us assume the function $$f(t)=\displaystyle\frac{t}{1+t}$$. Differentiating with respect to $$t$$, we get:

$\frac{df(t)}{dt}=\frac{1}{1+t} - \frac{t}{ {(1+t)}^2} \\ = \frac{1}{ {(1+t)}^2}$

This shows that $$f(t)$$ is monotonically increasing. A function $$f(x)$$ is monotonically increasing if for $$x_1 \leq x_2$$, we have $$f(x_1) \leq f(x_2)$$. Since our $$f(t)$$ is monotonically increasing, we can write that for $$t_1 \leq t_2$$:

$$$f(t_1) \leq f(t_2) \\ \Rightarrow \frac{t_1}{1+t_1} \leq \frac{t_2}{1+t_2} \label{eq:1}$$$

Set $$t_1=d(x,y)$$ and $$t_2=d(x,z) + d(z,y)$$. We can immediately see that $$t_1 \leq t_2$$. Thus substituting these values into $$\eqref{eq:1}$$, we get:

$$$\frac{d(x,y)}{1+d(x,y)} \leq \frac{d(x,z) + d(z,y)}{1+d(x,z) + d(z,y)} \\ = \frac{d(x,z)}{1+d(x,z) + d(z,y)} + \frac{d(z,y)}{1+d(x,z) + d(z,y)} \label{eq:2}$$$

We see that:

$\displaystyle\frac{d(x,z)}{1+d(x,z) + d(z,y)} \leq \frac{d(x,z)}{1+d(x,z)} \\ \displaystyle\frac{d(z,y)}{1+d(x,z) + d(z,y)} \leq \frac{d(z,y)}{1+d(z,y)}$

Thus we can rewrite the above inequality in $$\eqref{eq:2}$$ as:

$\frac{d(x,y)}{1+d(x,y)} \leq \frac{d(x,z)}{1+d(x,z)} + \frac{d(z,y)}{1+d(z,y)} \\ \Rightarrow \bar{d(x,y)} \leq \bar{d}(x,z) + \bar{d}(z,y)$

thus, proving the Triangle Inequality.

This is the central idea in proving that the distance metric in a sequence space of all bounded and unbounded complex numbers (Kreyszig 1.2-1) has a metric defined by:

$d(x,y)=\displaystyle\sum_{j=1}^\infty \frac{1}{2^j} \frac{\vert \zeta_j - \eta_j\vert}{1 + \vert \zeta_j - \eta_j\vert}$

where $$x=(\eta_j)$$ and $$y=(\zeta_j)$$.

tags: Mathematics - Proof - Functional Analysis - Pure Mathematics - Kreyszig