Definition talk:Right Shift Operator

What happens when the underlying set of the domain of $R$ does not contain an element $0$? --prime mover (talk) 21:12, 18 February 2021 (UTC)

I don't claim that this is the most general case. Clearly, this can be extended to a sequence of elements of any number field. But, to my knowledge, everybody defines this with a zero appearing as the first element. Since a field is defined to have a zero, this seems to be a valid way to define it so. At this point I expect this to be just a convention (both 0 and 1 are quite innocuous in their own ways), but maybe there is another reason why it cannot be any finite element. Most likely determined by the way how norms are handled.--Julius (talk) 21:44, 18 February 2021 (UTC)

Does it then in fact need to be specified that the underlying set has to be of a certain type? For example, does it have to be a field? Sorry to fuss like this, but coming as I do from an abstract algebraic perspective, I have learned that it is important to properly specify the domain. --prime mover (talk) 22:55, 18 February 2021 (UTC)

The most detailed requirement that I could find is that $X \in \set {c_0, c, \ell^p}$ which are zero-limit, convergent or $p$- sequence spaces and they are requirements on sizes of their elements. Now, whether something has to be a "field" or "topological vector space" or something else depends on where you look, but at least these definitions apply to $\C^\N$. Anything beyond that is beyond my expertise or needs.--Julius (talk) 07:32, 19 February 2021 (UTC)

Bah, I hate source works which don't define their terms fully. It's all predicate on what Sequence Space is defined as, but, as you say, if it is not defined properly in Sasane, we are just going to have to wait till we find a source which does it properly. --prime mover (talk) 08:36, 19 February 2021 (UTC)

That may take a while. I ran through over a dozen of other books and majority of them stick to $R: \ell^2 \to \ell^2$. A few others use $\ell^p$, one even supplied a norm. I also saw something lik $\mathbb F^\N$, which is then immediately followed by "where $\mathbb F$ denotes the field of real or complex numbers". I think we should simply choose the most general case WE can handle and use it throughout our work.--Julius (talk) 10:08, 19 February 2021 (UTC)

I disagree with "choose the most general case ..." My approoach is that we pick what's in the source we are processing and use that, then as and when we find something which applies to a more general object, we set up a transclusion page to put that more general definition in. Otherwise we have simple concepts like what I've just been doing, gradient and divergence, part of the cornerstone of undergraduate applied mathematics, expressed in terms that not only are completely incomprehensible to the undergraduate, but also not even defined in $\mathsf{Pr} \infty \mathsf{fWiki}$. As a consequence we get a reputation for intellectual elitism. I'd like to avoid that if at all possible. --prime mover (talk) 10:37, 19 February 2021 (UTC)

"Most general case" was meant for exactly this question since it was not clear whether we should go beyond $\C^\N$. I also added "case WE can handle" and I stated that $\C^\N$ is the most I and the majority of other books do handle. If complex numbers are elitist, we can always provide options for reals, rationals and so on. The problem is that on one hand, we are asking for rigorous extensive definitions, possibly beyond the coverage of our sources, while on the other hand, we complain that they are too involved and detached from the common man. Most people dealing with divergence or curl are content with their standard definitions, yet from the point of manifolds, standard definitions are lacking statements about their structures which are irrelevant in the undergraduate world. Of course I want this to be accsessible to everyone. That is why I have been prooving and reprooving results of metric spaces for normed vector spaces, because for some people metric is elitist. I would love to have a better source, but there is only so much time to look in the dark, so please bear with me when the domain is not fully defined. This was written by someone more competenet than me and aimed at students, so plausibly it was satisfactory.--Julius (talk) 11:24, 19 February 2021 (UTC)

Metric spaces is the most basic generalisation of the Cartesian real number spaces of all, and occur every time you turn round. It is far from "elitist" to start discussing them. Normed vector spaces, on the other hand, were not even mentioned in my degree course, so can hardly be "more accessible" and "less elitist" than metric spaces.

All normed vector spaces are metric spaces but the converse is not true, so what is less abstract? I did not encounter a metric well into my graduate courses. A vector space with Euclidean norm, on the other hand, was mentioned within the first semester of undergraduate classes (without referring to NVS). Probably because I began with engineering studies.

But my own involvement on this site is becoming more and more peripheral. It's only ever been something I do for fun. I have lost track of the direction we are going on $\mathsf{Pr} \infty \mathsf{fWiki}$ and every day I think I have picked up the thread again I find out I'm wrong. I'll let you take the lead, as you have a better idea of the current strategy and direction of the site than me. --prime mover (talk) 00:29, 20 February 2021 (UTC)

I think you are too humble. You have been handling this for years and are probably responsible for more than 90% of what we have here. This is the opposite of a peripheral contribution. Although there are some gaps (probably dictated by interest and competence), we are slowly filling them. So far your work on vector analysis looks great, and I believe that in the process a few other topics will become more accessible, which will allow us to pick these low hanging fruits.--Julius (talk) 12:56, 20 February 2021 (UTC)

I'll carry on doing my best, but I am still having trouble finding the sweet spot between "as general as can be" against "in an accessible context for beginners".

My own engineering degree was in electronics, so we never got beyond the $3$-d space applications, in which all the relevant vector analysis results seemed sufficient at the time to explore as much of the field as necessary to provide a solid grounding, but the general case of general vector spaces was not even touched upon. Then in my topology module in my later MMath we did metric spaces from the abstract viewpoint, touching down to ground only at certain specific points which were far removed from direct applications to physical applications. Even my fluid mechanics courses only ever went as far as 3-d vector spaces. So the abstract concept of a vector space, and indeed a normed vector space, has only ever been presented to me as an interesting generalisation in the abstract plane, accessed via my own subsequent studies in the haphazard accumulation of texts that I have gathered from various sources since then. I will try to get up to speed. --prime mover (talk) 13:26, 20 February 2021 (UTC)