Talk:Riesz–Thorin theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Constraints on the order of the indices?[edit]

The statement of the theorem seems to place undue restriction to the order of the indices. For example I want to choose p0=1, q0=∞, and p1=2, q1=2, but the theorem (as it's written here) seems to want me to ensure p0≤p1, and q0≤q1.

The choice of indices above is used in the motivation section for the Fourier transform, so the theorem should work for this case. — Preceding unsigned comment added by 86.169.149.174 (talk) 16:48, 8 May 2014 (UTC)

ambiguous sentence "Its usefulness stems from ..."[edit]

What exactly does this mean: "Its usefulness stems from the fact that some of these spaces have much simpler structure than others (namely, L2 which is a Hilbert space, L1 and L∞). " ? It it not clear whether the "namely ... L1 and L∞" part applies to "much simpler" or "others". I suppose L1 and L∞ are simpler but from the sentence, I can't tell for sure. For L2, the qualificative "is a Hilbert space" helps disambiguate. Much less so for L1 and L∞. Do L1 and L∞ also have simpler structure than the other Lp ? (Or is it just a matter of having a "simpler" metric ?) --FvdP 19:35, 29 Oct 2004 (UTC)

Of course they have a more complicated structure :-( Is the new formulation better? Gadykozma 20:09, 29 Oct 2004 (UTC)
Not much (IMO...). If you really mean that L1 and L(inf) are examples of the "other" = "more complex" kind of algebras, it's better to tell it explicitly. If your mention of L1 and L(inf) is unrelated to the previous sentence, why not something like: "The spaces involved are often L2 (which is a Hilbert space), L1 and L^inf."" --FvdP 20:22, 29 Oct 2004 (UTC)
No, it would be incorrect to say that they are simpler, they are not. Particularly is a non-separable behemoth. Nonetheless, sometimes they are easier to analyze, as in the examples. I guess "simple" is not so simple... Gadykozma 20:41, 29 Oct 2004 (UTC)

BTW, just wondering: In your example I see theorems that link L^p and L^q where 1/p + 1/q = 1. So when p=q you get theorems about L^2 alone. Is this related to L^2 being simpler and/or Hilbert ? (I know I am probably only exposing my utter ignorance of the matter by asking this question ;-) --FvdP 20:28, 29 Oct 2004 (UTC)

Yes, it's almost the same. is 's dual and a Hilbert space is a self-dual Banach space. Gadykozma 20:41, 29 Oct 2004 (UTC)

Constraints on p and q?[edit]

My question is what are the constraints on p and q in the theorem. Can they be 1 or infinity in addition to anything between, or just in between, or 1 but not infinity. This is because the theorem assumes L^p and L^q have a common dense subspace as domain, but on an infinite measure space if p=infinity and q<infinity then no subspace of L^q is dense in L^p, hence the assumptions of the theorem cannot be satisfied, it seems to me. Scineram (talk) 17:50, 19 May 2008 (UTC)

Yes the cases 1 and ∞ are included. To be a bit more formal the operator doesn't need to be defined on a subspace of Lp and Lq, but rather on a space (say D) so that is dense in Lp and is dense in Lq. There are a few mild assumptions about D, it should contain characteristic functions of finite measure sets and should contain truncations. Thenub314 (talk) 09:54, 10 October 2008 (UTC)

Generality of Theorem[edit]

"This theorem bounds the norms of linear maps acting between Lp spaces." - This is true, but not the entirety of the theorem and very misleading. The theorem is actually proved from one linear space, D, of measurable functions on a general measure space to another space of measurable functions on a general measure space as long as D contains all characteristic functions and truncations. Grimtageuk (talk) 14:45, 12 December 2012 (UTC)

Mistake log-convexity of norm[edit]

I think there is a mistake here. I tried in vain to prove the statement " p ↦ log || f ||p is convex " but, in fact, I think is the map 1p ↦ log || f ||p to be convex (see Lemma 2 here --Gim²y (talk) 17:12, 27 July 2014 (UTC)

Definition of "operator norm"[edit]

Wouldn't it be helpful to include the definition of ? I presume it is the usual one, i.e.