Talk:Riesz–Thorin theorem

Valid only over the complex field?

It is unclear whether the statement of Riesz--Thorin theorem is valid for spaces over the real field. The article says near the bottom that "the complex-analytic nature of the proof of the Riesz–Thorin interpolation theorem forces the scalar field to be C" but this restriction doesn't appear in the statement of the theorem. Is the statement true over R?

Constraints on the order of the indices?

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)

I don't see how the ordering affects the Fourier transform statements. The ordering it is used though in the layer-cake decomposition. The restriction however is without loss of generality. To lift the restriction we would have to use L^min{p0,p1} and L^max{p0,p1} in the layer-cake decomposition, still the Proposition itself is order agnostic. Did I miss anything? — Preceding unsigned comment added by 192.16.197.196 (talk) 14:19, 15 December 2017 (UTC)

ambiguous sentence "Its usefulness stems from ..."

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 $L^{\infty }$ 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. $L^{p}$ is $L^{q}$'s dual and a Hilbert space is a self-dual Banach space. Gadykozma 20:41, 29 Oct 2004 (UTC)

Constraints on p and q?

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 L^p and L^q, but rather on a space (say D) so that $D\cap L^{p}$ is dense in L^p and $D\cap L^{q}$ is dense in L^q. 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

"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 $L^{p}$ norm

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 1⁄p ↦ log || f ||_p to be convex (see Lemma 2 here --Gim²y (talk) 17:12, 27 July 2014 (UTC)

Definition of "operator norm"

Wouldn't it be helpful to include the definition of $\|T\|_{L^{p}\rightarrow L^{q}}$? I presume it is the usual one, i.e.

$\|T\|_{L^{p}\rightarrow L^{q}}:=\sup _{f:\|f\|_{p}=1}\|Tf\|_{q}.$