From: Stephen Montgomery-Smith Subject: Campaign to rename L_p spaces Date: Fri, 08 Sep 2000 04:47:51 GMT Newsgroups: sci.math Summary: [missing] I feel that the L_p spaces are misnamed. To recall the definition, if (Omega,F,mu) is a sigma-finite measure space, then L_p(mu) consists of those functions (or rather equivalence classes of functions under the relation equal almost everywhere) such that ||f||_p = ( int_Omega |f|^p dmu )^(1/p) is finite. I feel that this space is misnamed, and should actually be called L_{1/p}. So for example, L_3 should really be called L_{1/3}. Why? Well consider for example Holder's inequality: int fg dmu \le ||f||_p ||g||_q where 1/p + 1/q = 1. This relation is really quite complicated. It can be stated several ways (p-1)(q-1) = 1 or q = p/(p-1). If I tell you p=3, you think a bit, and realise q=3/2. Now with the new naming convention the relationship is much simpler: p+q = 1. If p=1/3, without thinking you see q=2/3. OK, so there is a slightly complicated relationship between p and q: we can live with that. But now consider interpolation: suppose T is a linear function that maps L_p0 to L_q0 and L_p1 to L_q1. The Reisz-Thorin interpolation theorem tells us that T also maps L_p to L_q for any 0<=theta<=1 where 1/p = (1-theta)/p0 + theta/p1 1/q = (1-theta)/q0 + theta/q1 OK, quick: if T maps L_2 to L_4, and L_5 to L_8, what does it map L_3 to? With the new naming scheme, the relations are much simpler p = (1-theta) p0 + theta p1 q = (1-theta) q0 + theta q1 A more compelling example - the Sobolev inequalities. This tells us that for f:R^d->R with Lebesque measure, then ||I_alpha f||_p <= c ||f||_q where I_alpha is the alpha fractional integral, and 1/p + d/alpha = 1/q (1=0). So in three dimensions where does I_1 map L_2? Use the formula, or use the new naming scheme: I_1 is one integral spread over 3 dimensions, so it affects the index of 1/2 by substracting 1/3, leaving 1/6, so L_6. This may seem like a trivial issue, but I have seen accomplished professional mathematicians struggling to see which space is mapped into by a Sobolev inequality. It's painful to watch, just like watching mathematicians counting change. Indeed I am facing these issues trying to follow all the indices understanding restriction inequalities (as in Stein-Tomas), and the results concerning the Kakaya sets. I just find it hard to "feel" where these indices come from, and how they relate. I know that my job would be so much easier if this new naming scheme were adopted. I know that the history will hold back renaming L_2 to L_{1/2}, but it would be a big help. Well, that's off my chest. (Please don't take me too seriously.) -- Stephen Montgomery-Smith stephen@math.missouri.edu http://www.math.missouri.edu/~stephen ============================================================================== From: lrudolph@panix.com (Lee Rudolph) Subject: Re: Campaign to rename L_p spaces Date: 8 Sep 2000 19:46:15 -0400 Newsgroups: sci.math israel@math.ubc.ca (Robert Israel) writes: >In article <8pb66h$ggl$1@panix5.panix.com>, >Lee Rudolph wrote: > >>I suggested to Stephen in e-mail that, considering how simple >>TeX makes it to typeset the Polish crossed L, he might consider >>trying to establish $\L_p$ as the notation for $L_{1/p}$. (Sort >>of a conservation of slashes effect, as well as an hommage to >>Banach.) Heck, I'd do it, if I ever had occasion to write about >>such critters. > >Which might confuse lots of people, especially those with >less-than-perfect eyesight... >but how would you pronounce these critters? There ought to be some Polish speakers here. Biding their authoritative responses, I will generalize from what I understand to be a reasonable pronunciation of \Lukasiewicz, and suggest "whoopie spaces". Lee Rudolph