Kind of odd that this doesn't really go into metrics directly nor does it mention one of the more interesting cases which is infinite dimensionality.
In the context of dimensional analysis, what is "infinite dimensionality"? Keep your answer brief, only one or two short paragraphs.
In the context of dimensional analysis, the term "infinite dimensionality" typically doesn't refer to the physical dimensions (like length, time, and mass) that one might immediately think of. Instead, in mathematical and theoretical physics contexts, it often pertains to spaces, particularly function spaces, that have an infinite number of dimensions. For instance, the space of all possible functions of a particular type can be thought of as having infinite dimensionality, as each function can be uniquely described by an infinite series of coefficients.
One of the most commonly encountered infinite-dimensional spaces in physics is the Hilbert space, used in quantum mechanics. Here, the state of a quantum system can be represented by a vector in an infinite-dimensional space. While the idea of "infinite dimensionality" might sound abstract and counterintuitive, it provides the necessary mathematical framework to accurately describe and predict a variety of physical phenomena.
That concept of infinite dimensional meters per infinite dimensional seconds doesn't work. With infinite dimensions that the whole idea of metrics breaks down. This gets used as a tactic: If you want to present some basic spatial concept without defining and then being bound to a specific metric then you might use infinite dimensional spaces for that since they are free of traditionally considered metrics.
This is rather exotic territory. The vast majority of phenomena that people find interesting can be characterized using eight or fewer dimensions.
That being said, I did try to connect infinite-dimensional problems with dimensional analysis in this post. Towards the end of my section on dimensions with complex powers, I applied dimensional analysis to the study of periodic signals and ended up arguing that (complex) Fourier basis elements can be thought of as dimensionful objects. I haven't seen dimensions used like this before, so I was hoping to earn the vote from fans of functional analysis ;)