Talk:Closed Real Interval is Compact

Sorry, but if you want to put a proof forward that uses Normed Vector Spaces, then you are free to do so -- but the proof as given here, and as cited in the McCarty work, operated on $\R$ being a metric space.

I have reverted it, because that's what the McCarty citation gave. If you want a Normed Vector Space version, you need to write it as a separate proof and then you can add your appropriate citation to that proof. --prime mover (talk) 20:07, 8 November 2020 (UTC)

Sorry, wasn't quite what I had in mind.

Perhaps I'm working towards the impossible, but I was hoping that we could achieve these results without duplicating the entire thing.

The general idea is that a set is "compact" in the reals independently of whether it is viewed as a metric space, topological space or normed vector space or whatever other type of space we happen to be dealing with at the time. Then we don't have to define a compact real space multiple different ways, and merely the method of proof would be different.

Never mind, I expect I'll be told I'm too ambitious, and that it is impossible to present the same object viewed from different angles, and we're only ever going to be able to present them as completely different objects. --prime mover (talk) 17:08, 9 November 2020 (UTC)

On one hand, I am covering normed vector spaces, so I prefer having my proofs based on theorems and definitions of normed vector spaces. On the other hand, we already have cases like Derivative of Exponential Function with several different proofs, so why should we stop there? Of course, some results can be proven in several different levels of abstraction and taking this "elevator" up and down may seem superfluous. On the other hand, not everything generalizes that easily, and logging when this happens is not something that is reflected in literature very often. --Julius (talk) 17:30, 9 November 2020 (UTC)

The point is that compactness in the real number line is the same thing whether defined in terms of metric spaces or topological spaces or normed vector spaces. I think what we need to do is establish that in the real number line they mean the same thing.

In fact, what is the difference between a normed vector space and a metric space in the real number context in the first place? Can we establish that a metric space (with the standard Euclidean metric) *is* a sort of normed vector space (or vice versa) already? I believe the answer to that is "yes of course we can" in which case the real challenge of the day is establish which properties of either one are common to both, and under what circumstances they overlap. Then we can make an attempt to unify the whole mess rather than just have two separate parallel threads proving exactly the same things using what ultimately boil down to the same techniques.

I'm currently having the greatest of difficulties unifying the approach to vectors from two different approaches: abstract algebra and physics. The former does not go anywhere near the language of directions and magnitudes, and the latter stays more or less rigidly (or at least the elementary texts do) within the scope of a 3-d Euclidean space. Okay, while magnitude is covered well in the concept of norms (hence your normed vector space), direction is rarely mentioned from the abstract algebraic viewpoint, which even treats dimension as an emergent property to be considered as an afterthought. Hence in these circumstances to unify the approaches is a major headache which I am not doing a very good job of myself. --prime mover (talk) 20:56, 9 November 2020 (UTC)

I would also like to find a way to climb up and down the ladder of abstraction vertically, although I also like having horizontal proofs on multiple levels unless they are in essence a carbon copy of each other. What we know is that all normed vector spaces are metric spaces, and all metric spaces are topological spaces. The reverse direction in general is more restrictive. So if something can be proven at normed vector space level, then if an equivalent is needed in metric or topological case, then indeed a rather simple theorem would provide this connection. The problem arises when we find a result at a topological level and we need an equivalent at a lower level. For example, not all metric distance functions are induced by a norm, and so it starts depending on the context.

The second problem is familiar to me. In the context of General Relativity, where differential geometry and tensor calculus are the main tools, there are two approaches as well. From physicist's perspective, all vectors and tensors are written in index notation, which implies that a certain basis is implied. This is not the case in the mathematical approach, where almost nothing carries any index, and something is a vector as long as it satisfies the axioms of vector space. This is not surprising, because mathematicians are mostly interested in coordinate-independent results, so the basis does not matter. I am not sure whether a complete unification is needed. Maybe what we need is a patch connecting both pieces where it is needed. I believe that most questions can be answered by following a book of differential geometry and differential manifolds, or a properly mathematical book on physics like the one by Vladimir Arnold.--Julius (talk) 22:14, 9 November 2020 (UTC)

I have a vast shelf of books but I don't have the Arnold. All take a subtly different approach. I have been trying to approach all my sources in parallel but that's hard work, so I'm going to use the approach of running with one, then evaluating what's been put up in light of the other. But it's late, I'm getting old (no joke) and I'm losing my ability to concentrate. Tomorrow is another day. --prime mover (talk) 23:28, 9 November 2020 (UTC)