Talk:Euclidean Space is Banach Space

I'm hopelessly confused -- is or is not a Banach space a normed complete metric space or just a complete metric space? It has to be one or the other. --prime mover (talk) 14:33, 12 May 2020 (EDT)


 * Reverted back to normed vector space. I myself got confused because I saw cases e.g. Euclidean Space is Banach Space, where Complete Metric Space was concluded to be Banach Space. So I assumed that in some sense Banach space could be generalized. But since it is not the case (Banach space is subset of Complete Metric Space), I will relink articles I modified, and we have to check where else complete metric space jumps to conclusion. --Julius (talk) 16:08, 12 May 2020 (EDT)


 * Ok, I see that there is also Euclidean Space is Normed Vector Space. But then, strictly speaking, we need one more short step to show that completeness from metric space gets transferred to normed vector space. In general, I find these similarities between metric and normed spaces very slippery.--Julius (talk) 16:18, 12 May 2020 (EDT)


 * From having looked at it, this area seems already to be pretty complete: you have a complete metric space, which is what it is, and then you have a complete metric space which has a norm on it, which is then a complete normed metric space, but we don't call it that, we call it a Banach space. So we have Euclidean space is complete metric space as one result, and then when you establish that there is in fact a norm on a Euclidean space, you then prove it is a Banach space by first proving the norm is what it is, then you invoke the proof that a Euclidean space is a complete metric space. Job done, move on. --prime mover (talk) 17:05, 12 May 2020 (EDT)


 * I think the issue might be that the definition of a complete normed vector space does not explicitly relate to completeness in the induced metric, which it is. The definition of complete normed division ring and Cauchy sequence in normed division ring makes the connection explicit. --Leigh.Samphier (talk) 19:02, 12 May 2020 (EDT)


 * Oh right okay. Is there a way to make this explicit in the names of the pages? (Using the forward-slash technique in this context is not optimal. The only reason it should ever be used is for transclusion, for the obvious reason.)


 * As I see we have Definition:Complete Metric Space:


 * "A Definition:Metric Space $M = \struct {A, d}$ is complete every Definition:Cauchy Sequence (Metric Space) is Definition:Convergent Sequence (Metric Space)."


 * (presentation of links removed for clarity)


 * Then we have Definition:Complete Normed Vector Space:


 * "A Definition:Normed Vector Space $M = \struct{X, \norm {\, \cdot \,} }$ is complete every Definition:Cauchy Sequence in Normed Vector Space is Definition:Convergent Sequence in Normed Vector Space."


 * Then we have Definition:Banach Space:


 * A Banach space is a Definition:Complete Normed Vector Space Definition:Normed Vector Space


 * ... so unless there has been some inaccuracy in determining which notion of "complete" is used in any of these different circumstances, I can't see the difference between Definition:Banach Space and Definition:Complete Normed Vector Space.


 * If there are subtle but serious differences between these different constructs, then we need to highlight what these differences are (that is, by presenting one or more examples illustrating those differences, probably best), and if they are ultimately the same in all circumstances, then the apparently-different definitions and other apparent differences need to be gathered in one place like a definition page with multiple defs along with a page proving their logical equivalence.


 * This is an area of mathematics I am not clear about. Every time I think I have a grip on an aspect of it, something (like the above perceived "slipperiness") confuses me. The hope is that can cut through this confusion. So I can't help here apart from ask stupid questions. --prime mover (talk) 02:30, 13 May 2020 (EDT)


 * Banach space is complete normed vector space and vice versa. Complete metric space in general is not Banach space. In some books Banach space is not used as the main name, so probably I defined it as I found in my book, and only later learned the standard here. I can relink all articles and absorb this through a redirect.


 * Then there is the concept of a complete metric space with a norm, which could be a "complete normed metric space"? One of the things which has confused me is the fact that my treacherous eyes are skimming over these definitions and not taking note where at one point it says "metric" and at another point it says "vector".


 * The approach by Definition:Complete Normed Division Ring is much better, because it hosts both definitions. My question is, whether it is enough just to define it in two ways, or we actually have to prove equivalence of definitions. And again this slipperiness of similarities of vector space and division ring. It seems that for some notions there are bundles of abstractions, and then for a theorem with both endpoints in lower level of abstraction one can either stay in there, or appropriately induce a higher level of abstraction, do something there and then come back down.--Julius (talk) 02:47, 13 May 2020 (EDT)


 * IMO if we have two different approaches, we gather those approaches into one page and prove equivalence.
 * None of this is made any easier by the "intuitive" approach to vectors taken by elementary texts on physics and applied maths where a vector is defined as a "line in space". Going up to the abstract algebraic definition is seriously scary, and it would be a serious public service if we can then provide not only a rigorous, but also an intuitive, explanation of how a "vector field" (as used, for example, in electromagnetic theory) is the "same thing" as a vector space as defined abstract algebraically. (My lack of progress on much in the way of physics is caused by my being daunted by how to frame the arguments in a rigorously mathematical way rather than relying upon an intuitional understanding of visualising arrows in space.)
 * After that, of course, we then have to explore the theory of spinors. --prime mover (talk) 03:18, 13 May 2020 (EDT)


 * Indeed, what is a vector in physics is not a vector in mathematics. Most often physicists have components of a vector in mind instead of an element of a vector space. There are a lot of good books about physics or even mathematical tools in physics that would not fit the style here. Then there is quantum field theory with particle physics, where a lot of notions are ill-defined in mathematical sense, but that does not prohibit physicists "proving" theorems and matching not-entirely mathematical predictions with experiments. But this is really beyond our scope. Also establishing actual axioms of physics (Laws of Newton vs Stationary action) is a headache. That's why it is one of Hilbert's problems. We should approach physics as a two-headed monster: mathematical and intuitive-theoretical.


 * Just a minute, I need to get this straight (I have this on my plate at this very moment). "what is a vector in physics is not a vector in mathematics" Truly? In what ways are they qualitatively different? I know they are defined and denoted in a different way using different approaches -- but at base, is not a vector in physics exactly the same as a vector in mathematics? Apart from the fact that in physics there is often a notion of applying that vector to a specific point (that is, along a specific line in space), which is addressed by the concept of a directed line segment, there seems to me that there is in fact no actual difference between the concepts at all.


 * This is of the utmost importance in e.g. electrostatic field theory, where an electric field is exactly a vector space, and the force acting on an electric charge is exactly a vector. What am I missing? --prime mover (talk) 05:37, 13 May 2020 (EDT)


 * I would like to imagine that (ultimately) this is actually not beyond our scope. Yes I know it's early days yet, we still need to work on a) making the mathematical groundwork rigorous and b) fleshing out the physical definitions (work in progress, sporadically), but I have seriously high hopes of at least getting Maxwell's equations and Schroedinger's Wave Equation handled (and the transcendental beauty of the Hermite polynomials displayed in all their glory complete with application), and I even entertain notions of proving Aufbau (IMO the most profound scientific truth yet discovered). --prime mover (talk) 04:50, 13 May 2020 (EDT)


 * At the base of intuition it is the same object. This will become more apparent, when we start working on differential geometry. There very frequently you start with a vector $X$ in some basis $e$ such that $X = X^\mu e_\mu$, and then you drop $e_\mu$. So now the language is simplified to "$X^\mu$ is a vector", which is really a component wrt a certain basis vector wrt a certain basis. The problem is more pronounced, when one starts with commuting basis and switches to a noncommuting one. It can also be that we have to write down a similar maths-physics dictionary (and these things actually exist. Here is a humorous digest https://imgur.com/BLl1G0U).--Julius (talk) 06:06, 13 May 2020 (EDT)


 * Most classical theories are mathematically solid, as well quantum mechanics and statistical mechanics (with some care). For them it's a matter of question of how we agree to present them here.--Julius (talk) 06:06, 13 May 2020 (EDT)


 * As for abstractions, there are also peculiarities within well-defined notions like Definition:Convergent Sequence. They look very similar at first, but they are not the same and how they inherit properties from each other is not always well stated.--Julius (talk) 04:24, 13 May 2020 (EDT)


 * That's where work is needed, then? --prime mover (talk) 04:50, 13 May 2020 (EDT)