Could somebody please offer some intuition into how pseudo-distances work?
Something like what the geometric interpretation is, and how they differ from distances, would be appreciated.
Background: I am currently studying the so-called $p$-variation, which is defined as follows:
$$ \| X\|_{p, [0,T]}:=\sup_{ D \subset [0,T] }{\left( \sum^{|D|}_{j=1} | X_{t_j} - X_{t_{j-1}}|^p \right)^{\frac{1}{p}}}$$,
where $ X_{(\cdot)}: [0,T] \to V $ is continuous, $T$ is a real number, $V$ is a Banach space, $|\cdot|$ is the norm on $ V$, and $ D$ is a partition of $[0,T]$. Since the 1-variation coincides with the definition of arc-length, the $p$-variation is a generalisation of arc-length. Defining $ \mathcal{V}^p([0,T], V)$ as the space of $V$-valued paths defined on $[0,T]$ of finite $p$-variation, it turns out that $\|\cdot\|_{p,J}$ defines a semi-norm on $ \mathcal{V}^p([0,T], V)$, which hence induces a pseudo-distance on this space ($\|\cdot\|_{\mathcal{V}^p([0,T], V)} := \|\cdot\|_{p,J} + \|\cdot\|_{\infty,J}$ defines a norm, which induces a distance on $ \mathcal{V}^p([0,T], V)$ ). It would be interesting to understand the geometric intuition of what semi-norms are, to better understand the geometric intuition of two paths being close in $p$-variation pseudo-distance.
Geometrically speaking, a semi-norm is the same as a usual norm, but there may be points that are distinct but have distance $0$.
The $p$-variation semi-norm is probably not the first semi-norm that you have encountered. The usual construction of the Lebesgue spaces is defining the set
$$ \mathcal{L}^p := \{ f: \Omega \rightarrow \mathbb{R}: f ~ \text{measurable}, \int_{\Omega} \vert f \vert^p d \mu < \infty \} $$
equipped with the semi-norm
$$ \Vert f \Vert_{p} = \left( \int_{\Omega} \vert f \vert^p d \mu\right)^{\frac{1}{p}}$$
However, this of course means that if two measurable functions are equal $\mu$-a.s., then their difference has norm $0$. In order to solve this one passes to $\mathcal{L}^p / \ker( \Vert \cdot \Vert_p )$ on which $\Vert \cdot \Vert_p$ is a bona fide norm.
For the $p$-variation semi-norm we have $\Vert \alpha \Vert_p = 0$ for every constant $\alpha \in \mathbb{R}$. Hence the $p$-variation cannot distinguish constants and thus the distance between any two functions which differ only by a constant is $0$. To solve this one adds the term $\Vert \cdot \Vert_{\infty}$, which can distinguish constant functions from $0$.