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 . 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)