User:Lord Farin/Sandbox

This page exists for me to be able to test out features I am developing. Also, incomplete proofs may appear here.

Feel free to comment.

Over time, stuff may move to User:Lord_Farin/Sandbox/Archive.

Pointwise Operations on Mappings
Shortly, I will need a lot of pointwise operations on mappings, so much that I feel there is a case for a page 'Pointwise Operations on Mappings' or something similar. A quick pick (for (extended) real-valued fns):
 * $f+g$, $f g$, $\max(f,g)$, $\sup_i f_i$, $\limsup_i f_i$, $\lim_{i\to\infty}f_i$

and also notions like pointwise limit of mappings $\lim_j f_j = f$

It feels crafted and is unpleasant to keep introducing these every time they are used.

Maybe I could write pages for each of them (specifically for (extended) real-valued functions), and bundle these on 'pointwise operations on real-valued mappings' or something similar. I hesitate a bit because with my natural ability for abstraction, I don't need these myself; they are adequately formulated on Operation Induced on Set of Mappings for anyone capable of the required abstraction. It's just that a (relatively) self-contained treatment (of eg. analysis or measure theory) should not need to delve into the intricacies of abstract algebra when such isn't necessary.

More pages of this sort could be created, and put on ref'd abstract algebra page as examples. --Lord_Farin 07:51, 5 April 2012 (EDT)
 * That is, I would like the reader to comment on this (if he has an opinion). Go ahead, it's free :) --Lord_Farin 16:54, 5 April 2012 (EDT)


 * I would suggest a subpage of Definition:Operation Induced on Set of Mappings which specifically discusses the case where $S$ and $T$ are Definition:Real Functions (or whatever) and none of the baggage of the "set of all mappings" etc., just say $f$ and $g$ are mappings, $f \oplus g (x) = f (x) \oplus g (x)$ where $\oplus$ (or whatever symbol you use) is any operation: "Examples: plus, times, max, sup, etc.". Possibly a different page (or even a different subpage expressing the general result) for the general multifunction $\sup_i f_i$, $\limsup_i f_i$ etc., but I'm not sure how this would be crafted. --prime mover 17:46, 5 April 2012 (EDT)

Naming problems
Another one: Schilling introduces the following very convenient shorthands:
 * $\{u \le v\} := \{x \in \R: u (x) \le v (x)\}$

and similar, about anything you can think of ($\{u = \lambda\}:= \{x\in\R: u(x)=\lambda\}$ and so on). But how to incorporate this into PW? It is really very suitable, eg when writing $\int_{\{u\le v\}}d\mu$ and $\chi_{\{u\le v\}}$, paramount in measure theory. --Lord_Farin 17:33, 3 April 2012 (EDT)


 * Just thought that it would be best to introduce this on Set Definition by Predicate (which, on a side note, needs to be brought up to our evolved standards). Any comments? --Lord_Farin 07:56, 5 April 2012 (EDT)