Talk:Lp space/Archive 1

Minor edit

Hi, I turned five-or-so references to "Banach space"s into "wiki-hyperlinks" (they were plain text).

BTW, I noticed you refer to Sobolev spaces; you may (or may not) want to add as another reference the brand new book "Selected Works of S.L. Sobolev", ISBN 038734148X.

-- Biscay 10:33, 23 August 2007 (UTC).

Hi Biscay, it is customary to not link the same word repeatedly but only the first instance. Furthermore it is customary to add new entries to the bottom of talk pages. The + at the top of talk pages can also be used to add new sections to the bottom. --MarSch 11:57, 4 October 2007 (UTC)

The page has a lot of information about l^p spaces, but does not contain a definition! —Preceding unsigned comment added by 212.143.119.137 (talk) 10:47, 21 May 2009 (UTC)

Local convexity

Wait, I thought the problem with L^p spaces, 0<p<1, was that they weren't locally convex.

Also, who in the world decided to put the p in the subscript? When you have spaces with multiple indices like Sobolev spaces, I can understand the p landing somewhere odd, but there's only one index here, and I've never seen it downstairs... Loisel 20:49 Jan 28, 2003 (UTC)

I think you're right about the notation and we should change it; I'm not sure about the locally convex part though. Can you find it in some book? AxelBoldt 01:34 Jan 29, 2003 (UTC)

Several. Wojstaszczyk Banach Spaces for Analysts, page 9 or Rudin's Functional Analysis, page 37. By the way, I must contradict my note above now, I do know an author who puts the p downstairs. Wojstaszczyk does, but he's the only one I know. Sorry for the long delay, I never noticed your reply. Loisel 07:14, 27 Jun 2004 (UTC)

The reason you put p in the subscript can be that you write Sobolev spaces with p in the subscript, and these two p's are "things of the same kind".

Timur

Yeah, perhaps there should be a bit more emphasis here? The Sobolev space can be written, for example, W_p^{k,k'}, for some intersection of k and k' - differentiable spaces. In this case, the L^p is written L_p for consistency in the notation. The only point being, these notations are not all that rare. Cypa 06:06, 22 March 2006 (UTC)

Actually, putting the p in the subscript is a very common convention in Banach space theory; see for example several of the articles in the recent "Handbook of the Geometry of Banach Spaces" edited by Johnson and Lindenstrauss. Nonspecialists in Banach space theory (including most authors of functional analysis texts) almost always put the p as a superscript. For different people there are different reasons (besides habit and the reasons mentioned above) for preferring one convention or the other. The subscript emphasizes the relationship with the function of "raising to the pth power"; there are spaces related to more complicated functions which are sometimes analogously denoted, for example, L log L. On the other hand, when working with finite dimensional spaces, we are accustomed to seeing the dimension as a superscript, so it is convenient to put the p as a subscript to avoid confusing it with the dimension. 76.190.215.240 (talk) 16:59, 14 February 2008 (UTC)

Dual of L-infinity

Does anyone have any more to add about the dual of L^infinity? Lupin 15:21, 7 Feb 2004 (UTC)

I think it should be fairly easy to prove that the natural injection of L^1 into (L^infinity)** is surjective iff both of them are finite-dimensional (i.e. X is essentially a direct sum of a finite number of atoms of finite measure).

Prumpf 11:05, 10 Feb 2004 (UTC)

The injection of L^1 into (L^infinity)* (not **) is clearly surjective in the finite dimensional case. Let the measure space S be sigma-finite. Then L^infinity is not separable. If a dual space A* is separable, then A must also be separable. Hence L^1, which is separable, is not the dual of L^infinity.

However, I think there's a problem. In Banach Spaces for Analysts, by Wojtaszczyk, 1991, page 10: "The space L_infinity(Omega,mu)^* is (unless (Omega,mu) consists of a finite number of atoms) much bigger than L_1(Omega,mu) and is impossible to describe explicitly" (emphasis added.) I concur.

Loisel 05:03, 27 Jun 2004 (UTC)

What's impossible to describe explicitly supposed to mean, anyway? The Lp spaces are impossible to describe explicitly, in a way, so there doesn't seem to be any actual information added to the article by that phrase.

Also, your proof works in the sigma-finite case only, and thus it's slightly misleading to include the proof outline in the article. However, the general proof can be reduced to the sigma-finite case fairly easily.

Prumpf 17:31, 28 Jun 2004 (UTC)

I don't see how the general case reduces to the sigma-finite case -- are you sure of that statement? I think the proof I gave does generalize, if you replace "sigma-finite" by the appropriate notion of size for the measure space S. Say that S can be written as the disjoint union of E_a, sets of finite measure, with the index a running through an index set I. WLOG, I is of minimal size. Then, L^1 has a dense subset of cardinality #I; however, L^infinity does not (you can embed 2^I as a discrete set in L^infinity). If A* has a dense subset of a certain cardinality, then A must have a dense subset of the same cardinality. Hence we get that L^1 is not the dual of L^infinity.

I'm not quite sure what Wojtaszczyk means by "Impossible to describe explicitly." He could mean it in some formal sense (like what we mean of the Vitali set when we say it can't be constructed -- this is because it uses the axiom of choice in an essential way.) However, I suspect that's not what he means; rather, I suspect he's saying that nobody has given any good characterizations of the dual of L^infinity. I've seen it described as a space of distributions, but it's not the same space as the dual of C_c^infinity, nor is it the space of tempered distributions. Loisel 08:03, 29 Jun 2004 (UTC)

Not every measure space (not even every measure space in which there is no set $A$ with infinite measure that doesn't have a subset of nonzero finite measure; for those measure spaces it is trivial that the dual of L^infinity is larger than L^1) arises as a direct sum of finite measure spaces.

I think that if you assume that mu({x}) is finite for every x, the "trivial" proof I gave above works. If E={x|mu(x)=infinity} is not empty, I'm pretty sure you can split S into E and S\E and work from there. The proofs you give below appear kosher. Loisel 00:59, 30 Jun 2004 (UTC)

Uhm, the case in which your E isn't empty is rather simple. Still, E can be empty and I don't see how your proof would work, since that still doesn't mean S is a direct sum of finite measure spaces (trivially, consider R with infinity times the Lebesgue measure). Also, aren't there sigma-finite measure spaces for which L^1 and L^2 aren't separable (the product of infinitely many copies of the unit interval with the Lebesgue measure, for example)? Your proof seems to fail for those, I'm afraid. (And a fortiori, mine is then invalid as well, having used your proof) Prumpf 15:39, 30 Jun 2004 (UTC)

Wait, my proof is still valid, since it needs only the case where S is a countable sums of atoms, for which the separability argument holds. Prumpf 15:45, 30 Jun 2004 (UTC)

Yes, I see. Go for your proof. Loisel 19:28, 30 Jun 2004 (UTC)

Assume our measure space X is not sigma-finite, but that we can find a countable family {A_i} of disjoint measurable subsets with 0 < mu(A_i) < infinity (this covers all the cases that have been left out so far). Let Y be the measure space consisting of a countable numbers of atoms {y_i}, such that mu({y_i}) = mu(A_i). We can thus embed L^infinity(Y) in L^infinity(X) and L^1(Y) in L^1(X), and obtain a mapping (L^infinity(X))* -> (L^infinity(Y))* by contravariance. The diagram

L^1(Y) ----> (L^infinity(Y))*
  |                 ^
  v                 |
L^1(X) ----> (L^infinity(X))*

then commutes. Furthermore, the arrow in the right column is a surjection, by the Hahn-Banach theorem. Also, the arrow in the top row isn't a surjection. Assume the bottom arrow is. Choose y* in (L^infinity(Y))* that is not in the image of the top arrow, x* in (L^infinity(X))* which maps to y* under the right arrow, x in L^1(X) which maps to x*. We now modify x to be constant on the sets A_i by averaging over them, which results in a new x* but leaves y* unchanged. However, we can now choose y in L^1(Y) that maps to x, and maps to y* by commutativity, a contradiction.

Not as concise as I had hoped, but this really should work for nasty measure spaces (for which L^infinity presumably isn't the dual of L^1, but whatever) Prumpf 11:08, 29 Jun 2004 (UTC)

Slightly simpler: define a map L^1(X)->L^1(Y) by averaging over A_i, then prove the diagram

L^1(Y) ----> (L^infinity(Y))*
  ^                 ^
  |                 |
L^1(X) ----> (L^infinity(X))*

commutes. Assuming the bottom arrow is surjective, the diagram in fact looks like this

L^1(Y) ----> (L^infinity(Y))*
  ^                 ^
  ^                 ^
  |                 |
L^1(X) --->> (L^infinity(X))*

(this can all be done in Set), from which it follows that the top arrow was surjective as well, contradicting the result about sigma-finite measure spaces. Prumpf 11:40, 29 Jun 2004 (UTC)

I'm not sure I see the problem with identifying the dual of L^infinity with the ba space. Certainly, every linear functional phi: L^infinity(X) -> R gives rise to a finitely additive (signed) measure mu on X by defining mu(A) = phi(1_A). Since |mu(A)| = |phi(1_A)| <= |phi|, mu is bounded, and |phi| is just the variation of mu, the maximal integral of a measurable function f with |f(x)| = 1 for all x in X.

OTOH, if mu is a f.a. bounded signed measure, we see that there is exactly one linear functional corresponding to it (if

$f=\sum _{n\in \mathbb {Z} }2^{n}\times 1_{A_{n}}$, then $\phi (f)=\sum _{n\in \mathbb {Z} }2^{n}\times \mu (A_{n})$. It is bounded since mu is.

Looks good. Loisel 19:42, 30 Jun 2004 (UTC)

Thus, (L^infinity)* is isometrically isomorphic to the ba space, assuming it is defined to consist only of those measures which keep all of the original measure's null sets. That seems a fairly explicit description to me, if possibly not a very useful one.

Note that by the Radon-Nikodym theorem, L^1 can be identified with the subspace of the ba space consistion of all sigma-additive measures. Thus, the natural endomorphism L^1->(L^infinity)* is an isomorphism if and only if all finitely additive measures on X which keep all null sets are automatically sigma-finite, and it is trivial to see that it is exactly the finite-dimensional case where that is true. This is a much simpler proof than the one I've given above. Prumpf 16:02, 30 Jun 2004 (UTC)

Let's get our act together

In reference to the above, I vote we remove all lies from our math articles, now and forever, in the name of the great turtle A'tuin. Amen. Loisel 07:25, 27 Jun 2004 (UTC)

Topology

The section Further properties states:

The map sending f to ||f||_p is a quasi-norm, and L ^p is a quasi-Banach space, that is, a complete quasi-normed vector space.

This statement is true/confusing/contradictory with the rest of the article. Here's why: The Lp norm will assign a norm of zero for functions that are not identically zero; thus by definition, it is a semi-norm not a norm. Thus, "raw Lp space" is a Kolmogorov space topologically speaking (and that article has a good discussion of this). Only after the Kolmogorov quotient is taken, where we place in an equivalence class all functions having the same norm, do we get the "true" Banach space. So, somehow, the above sentence needs to be re-written and incorporated more directly into the discussion. linas 20:30, 10 December 2005 (UTC)

Is a quasi-norm the same as a semi-norm, i.e., something which satisfies all the axioms except that ||x|| = 0 does not imply x = 0? In that case, what's wrong with the discussion in the section L^p spaces? -- Jitse Niesen (talk) 20:57, 10 December 2005 (UTC)

Okay, I found out that a quasi-norm is something different. Now I don't understand Linas' comment at all :( As I understand it, "raw Lp space" is not a Kolmogorov space, because the functions which differ in one point are topologically indistinguishable. After taking the quotient, you get a Kolmogorov space. I have no idea what quasi-norms have to do with this, but I've never worked with p < 1. -- Jitse Niesen (talk) 21:20, 10 December 2005 (UTC)

Oh, well, I thought that quasi-norm is the same thing as semi-norm, since they both redirect to the same article (and that article doesn't otherwise define quasi-norm). My complaint wasn't about other discussions, it was about the quoted sentence in particular. Its not clear what it has to do with p < 1, if anything. I retract my complaint if we can clear up what a quasi-norm is.

Sorry for the confusing term "raw Lp space"; I was trying to refer to that thing that has the finer topology, "the space of functions that are p-integrable but has a topology that distinguishes functions that differ at a point." I don't know if that space has a name, or what the name would be. linas 21:44, 10 December 2005 (UTC)

Yes, the redirect from quasi-norm also misled me, so I removed it. A web search suggested that a quasi-norm satisfies ||x + y|| ≤ C(||x|| + ||y||) instead of the triangle equality. This makes more sense in the context. I also tried to clarify the article. -- Jitse Niesen (talk) 23:03, 10 December 2005 (UTC)

addition

Isn't it necessary in defining a vector space to define addition in that space?Loodog 04:27, 7 March 2006 (UTC)

Well, in Lp spaces, which are function spaces, the addition is just sum of functions. I guess nobody bothered to write that in the article, and it is rather clear from the context. Oleg Alexandrov (talk) 04:57, 7 March 2006 (UTC)

I would agree, but the function space article also fails to define addition (I think) on account of the definition of function space being too general. Also, citing vectors in Euclidean space as an Lp space in the article, it seems that the domain can be things other than functions. Maybe there should be a brief note saying, "Lp space is a function space. Addition in Lp space is the usual addition under any function space" or something like that.Loodog 20:41, 7 March 2006 (UTC)

Vectors are lp with a little "l", functions are Lp with a big "L". linas 00:53, 8 March 2006 (UTC)

The article isn't complete without a full definition of the vector space structure, so I've added the information you've requested. -lethe ^{talk +} 02:32, 8 March 2006 (UTC)

This page is a mess

This page is a mess -- "capital L"^p referes to Lebesgue spaces, not general normed vector spaces. And, the part on L^p in terms of Lebesgue starts with a measure space with some measure mu, then talks about Lebesgue integrability. What's going on? Please be more careful! —The preceding unsigned comment was added by 131.215.242.202 (talk • contribs) 04:22, 10 April 2006 (UTC2)

I guess the term "Lebesgue integral" can mean two things: 1. the supremum of simple functions less than the integrand over sets measurable with respect to any measure, or 2. the same supremum using the Lebesgue measure. If you understand the term in the first way, the more general way, there is no problem understanding Rⁿ with the l^p norm as a Lebesgue space (use the counting measure on the set n). There is also no problem with the phrase you are complaining about. I claim that we have to understand the term in its more general sense, at least for the scope of this article, since such spaces as L^p(M,μ), integrable over an aribtrary space with respect to an arbitrary measure, are found in the literature thusly denoted. This is the place to treat such spaces. -lethe ^{talk +} 21:27, 10 April 2006 (UTC)

Dual of L^1

Is it really necessary to assume that the measure is sigma-finite to conclude that "the dual of L¹(S) is isomorphic to L^∞(S)"? Theorem 2.34 of Adams, Sobolev Spaces, states that

$[L^{1}(\Omega )]'\cong L^{\infty }(\Omega )$

if Ω is an open, but not necessarily bounded, subset on R^n (the underlying measure is the Lebesgue measure). So the statement is true for more measures; what I'm wondering is whether it's true for all measures. I guess not, given the number of people that have contributed to this page. -- Jitse Niesen (talk) 09:38, 19 July 2006 (UTC)

Not true in general, unless you define L^\infty weirdly when \mu is not sigma-finite. I believe the correct definition would be {f|f restricted to E is bounded, for every sigma-finite E}. Note that the Lebesgue measure on R^n is sigma-finite, Adams's statement is a special case, not a generalization. Loisel 10:17, 19 July 2006 (UTC)

Oh dear, I forgot the difference between finite and sigma-finite measures. How embarrassing … Anyway, thanks for your quick answer. -- Jitse Niesen (talk) 10:39, 19 July 2006 (UTC)

Weighted Lp spaces

Sullivan, I understand what you're saying but it's just a question of notation, so I suggest it stays the way I put it now. If one understands that f d\mu is the measure such that f d \mu(E)=\int_E d d\mu then the notation is consistent.

Loisel, I understand "$f\,\mathrm {d} \mu$" notation. My comment that "notation is not definition" refers to the fact that a space is not defined by the notation used to describe it. The weighted $L^{p}$ space is defined to be a space of functions with certain properties. It is not defined to be $L^{p}(S,w\,\mathrm {d} \mu )$ or $L_{w}^{p}(S,\mu )$. These are just notations, and I deliberately included both in my revision of the article. Also, I must note that your revision itself used inconsistent notation for the space (different notation in body text and displayed equation)! Sullivan.t.j 16:12, 1 October 2006 (UTC)

I'm not sure if I'm understanding you right, but you shouldn't redo the L^p theory when you add a weight function; that's wasted effort. If \nu is the measure such that d\nu=f d\mu, then the weighted L^p space should be defined in terms of the already-defined space L^p(\nu). Otherwise, you have two possibly inconsistent definitions, and the reader may not be sure which theorems of L^p spaces apply to weighted L^p spaces.

If it's clear that a weighted L^p space is nothing more than an L^p space with a different measure, the reader understands that the theorems for L^p spaces and weighted L^p spaces are the same. Loisel 18:57, 1 October 2006 (UTC)

On further reflection, I agree that using $w\,\mathrm {d} \mu$ re-uses notation in a fairly nice way and saves space. Nonetheless, I think that it should be more explicitly stated: my first thought was that you meant something like $L^{p}(S,w,\mu )$ but had made a typo. So, my latest revision uses more notation but (hopefully) makes the reasoning more obvious. I have also fixed the subscript on the norm in the displayed equation (it used $L_{w}^{p}(S)$-notation). Sullivan.t.j 08:29, 2 October 2006 (UTC)

OK. Loisel 21:09, 2 October 2006 (UTC)

Minor comment

Under motivation it says "It turns out that this definition indeed satisfies the properties of a length function (or norm), which are that only the length of the zero vector is zero (snip)."

If I understand how length is defined in Minkowski space than 4-vectors of length zero exist that are not the zero vector. Maybe a better description of length function is needed? Unmasked 14:09, 8 October 2006 (UTC)

I have modified the wording of the offending paragraph to make it a little clearer. However, you are right to note that in Minkowski space there are non-zero vectors with zero "length"; the "length" function on Minkowski space is not a norm. Sullivan.t.j 16:26, 8 October 2006 (UTC)

L^1_2 space?

I seem to see things such as $L_{2}^{1}(X)=\{f:\int _{X}|f(x)|(1+x^{2})\,\mathrm {d} x<\infty \}$ a lot. Is that some variant of Lp spaces? (Is there some such thing as "L^p_q space", defined as something along the lines of $L_{q}^{p}(X)=\{f:\int _{X}|f(x)|^{p}(1+x^{q})\,\mathrm {d} x<\infty \}$?) -- 129.78.64.102 03:19, 31 May 2007 (UTC)

These are weighted Lp spaces, from what I know, with the weight being that function you multiply |f(x)| by. Oleg Alexandrov (talk) 03:40, 31 May 2007 (UTC)

Why "L"?

Why are these L spaces? Why that letter? 75.67.167.110 17:31, 26 July 2007 (UTC)

L is for Henri Lebesgue and his integral, which is the sense in which the integrals defining the L^p norms are defined. Sullivan.t.j 21:53, 26 July 2007 (UTC)

proof of completeness

I'd really love to have a good proof of completeness in the article instead of the "follows from convergence theorems" there is now. proof that L_p spaces are complete (at PlanetMath) might also help with accomplishing that. --MarSch 12:00, 4 October 2007 (UTC)

Dual or Antidual ?

In the properties of L^p spaces part, the functional G(f) is said to be in the dual of L^p, but it is in fact an antilinear map, so we should talk about the antidual of L^p and not its dual, if one want to use the map G.

Thus, we could say that the antidual of L^p is L^p' where p^-1 + p^-1 = 1, the antidual of L^1 is L^\infty and we recover the famous fact that L^2 is its own antidual, as any Hilbert space.

Its seems to me that if we use G( \bar f) instead of G(f) the previous properties still hold by replacing antiduality by duality, but one should keep in mind that if we consider Bochner spaces of functions with values in a complex Hilbert space, we only have G(f) at our disposal, since no \bar f is canonically defined. So the natural thing in the complex setting is antiduality, not duality. Please tell me if you agree or disagree.

HB. —Preceding unsigned comment added by 139.124.7.126 (talk) 06:15, 9 December 2007 (UTC)

You are correct that G (not G(f); that is a number) is antilinear in the complex case. I think you are also correct in saying that L^p is isomorphic to its antidual and anti-isomorphic to its dual in the complex case. We should probably just treat the complex case, or maybe use *-algebras and mention that the real numbers with the trivial involution trivially form a *-algebra and the complex numbers with conjugation also form a *-algebra.

I don't know much about Bochner spaces. Why can the involution of a complex Hilbert space not be pulled back to each Bochner space of functions to that Hilbert space? --MarSch (talk) 18:07, 9 December 2007 (UTC)

OK but it seems to me that there is no natural involution in an abstract complex Hilbert space, even if there is one in \C^n. So we have a lot possible involutions and none are canonical, so it is probably better to keep the antidual formulation. As you say, we can always recover the real case by noting that in this case antidual=dual. Another nice feature of antidual formulation is that the antiduality bracket is a natural extension of the scalar product of L^2, contrarily to the duality bracket. I don't know if these remarks are somewhere on Wikipedia, I guess it would be useful to mention them in this article.

139.124.7.126 (talk) 02:52, 10 December 2007 (UTC) HB

One can check that as p increases, the set l^p grows larger.

I think that this sentence is not correct: how can a set grow larger? TomyDuby (talk) 02:08, 24 July 2008 (UTC)

By getting in additional elements, I presume. Oleg Alexandrov (talk) 02:16, 24 July 2008 (UTC)

If $p<q$, then $\ell ^{p}\subsetneqq \ell ^{q}$. Loisel (talk) 02:22, 24 July 2008 (UTC)

Um, except that, for the special cases of sequences, one does have $\ell ^{1}\subset \ell ^{p}\subset \ell ^{q}\subset \ell ^{\infty }$ when $1<p<q<\infty$, and similarly for the set of measurable functions defined on the closed unit interval. This doesn't hold for general measurable functions on general measurable spaces, nor does it hold for measurable functions on open intervals of the real line. linas (talk) 18:29, 7 August 2008 (UTC)

Linas, I think this is incorrect. The sequence $a_{k}=1/k$ is in $\ell ^{p}$ for every p>1, but not in $\ell ^{1}$.

Huh? That just shows that $\ell ^{1}\subset \ell ^{p}$ is a proper subset. linas (talk) 23:05, 7 August 2008 (UTC)

On the other hand, for function, e.g., of the interval, you have that $1/{\sqrt {x}}$ is in $L^{p}(0,1)$ for every $p<2$, but not in $L^{p}$ for every $p\geq 2$.

Sorry, yes, right, for the capital-L's the order of inclusion is reversed, as your exercise from Rudin, below, points out. linas (talk) 23:05, 7 August 2008 (UTC)

I will now go through your edits carefully to see which of your criticisms are founded. Loisel (talk) 21:37, 7 August 2008 (UTC)

And so, in the section on "embeddings", I read

L^p(S) is not contained in L^q(S) iff S contains sets of arbitrarily small measure, and

which sure as heck does not seem to make sense to me, as, the closed unit interval has sets of arbitrarily small measure, and yet one has $L^{p}\subset L^{q}$ when $p<q$ for the closed unit interval! I mean, its been a while, and my memory is failing with age, but jeez, surely there's a bug here!? I removed the above sentence; someone please clarify the intended meaning! linas (talk) 18:45, 7 August 2008 (UTC)

Ohh, I see my mistake ... I reversed the order of inclusion, I switched around p and q. I guess the original statement was correct. My bad, please revert my edit. linas (talk) 23:10, 7 August 2008 (UTC)

All right, I see there was only one edit. In my copy of Rudin, Chapter 3, exercise 5 gives that $L^{s}(\mu )\subset L^{r}(\mu )$ if 0<r<s and $\mu (X)=1$. For the $\ell ^{p}$ spaces, see exercise 7. Loisel (talk) 21:53, 7 August 2008 (UTC)

You mean perhaps 1<r<s ? Since the metric is not a metric, its a pseudometric/quasimetric (doesn't obey triangle inequality or whatever, I forget) when p<1. What do you mean "I see only one edit"? Also, p.s. of interest might be Maharam's theorem which is what lead me to look at this again. linas (talk) 22:53, 7 August 2008 (UTC)

Rudin has 0<r<s, not 1<r<s. Loisel (talk) 00:44, 8 August 2008 (UTC)

It's a metric, but not a norm (doesn't satisfy the homogeneity condition). Anyway, I agree that there is something wrong about the statement in the article. It just can't be right as stated. siℓℓy rabbit (talk) 23:03, 7 August 2008 (UTC)

I'm running out the door now, I'm late and can't think straight; I do see that I reversed my p's and q's so perhaps the original statement was right after all. linas (talk) 23:12, 7 August 2008 (UTC)

(e/c)I retract the last part of the statement. The logic seems awkward to me. It seems that we should state it in a positive sense:

L^p(S) is contained in L^q(S) iff S does not contain sets of arbitrarily small measure,

Just a thought. (The unit interval contains sets of arbitrarily small measure, and so, for instance $L^{1}(0,1)\not \subseteq L^{\infty }(0,1)$.) siℓℓy rabbit (talk) 23:16, 7 August 2008 (UTC)

I've clarified it. Loisel (talk) 00:42, 8 August 2008 (UTC)

OK I'm drunk so I'm gonna stop editing. Check to see that I have the order right... Loisel (talk) 04:13, 8 August 2008 (UTC)

Looks much much better. One last request, then: present two special cases/lemmas, as these are the ones that commonly appear in many textbooks: the sequence, and the closed unit interval. Although these follow from the general case, I think visually, they stand out and would serve as a point of reference for anyone comparing this article to whatever else they're reading: For the special cases of sequences, one does have $\ell ^{1}\subset \ell ^{p}\subset \ell ^{q}\subset \ell ^{\infty }$ when $1<p<q<\infty$, and similarly for the set of measurable functions defined on the closed unit interval, one has $L^{\infty }([0,1])\subset L^{q}([0,1])\subset L^{p}([0,1])\subset L^{1}([0,1])$ when $1<p<q<\infty$. My apologies for having mislead/confused the issue above. linas (talk) 17:21, 8 August 2008 (UTC)

Superscript vs. Subscript?

Am I crazy? The few books/papers I have use subscripts, not superscripts: so L_p and not L^p, so it was jarring to see superscripts used in this article. Is superscript really the predominant practice today? Perhaps subscripts date back to an earlier era?linas (talk) 18:29, 7 August 2008 (UTC)

Many of the recent books use superscript (e.g. Rudin), but there are many others that use subscript. Loisel (talk) 21:37, 7 August 2008 (UTC)

modulus-linearly?

in the 2nd criterium of the definition of a norm given in this article we have: "the length of the vector changes (modulus-)linearly" what is the modulus part? i didn't learn this while taking graduate level analysis. if you go look at the norm article, it doesn't mention modulus anywhere on the page. lets either clarify in this article or link to something else that explains... or take it out. --watson (talk) 01:47, 30 October 2008 (UTC)

Open mapping unnecessary (in the section "properties of L^p")

In the description of the dual of L^p, the map κ is isometric and onto, so it is clear that the inverse exists, and is also isometric, hence continuous. I removed the invocation of the open mapping theorem.

There was a small inaccuracy in the discussion of reflexivity that I tried to fix: There is a map κ_p for every value of p, and two different values p and q were used in the diagram, with the same name.

Also added a word about the non-local convexity when p < 1. Bdmy (talk) 21:37, 2 November 2008 (UTC)

I like the revision of the first few paragraphs. We may want to rethink the placement of the 0<p<1 case, since the article now includes a section for that later on. siℓℓy rabbit (talk) 21:43, 2 November 2008 (UTC)

Case 0 < p < 1

The dual of $\ell ^{p}$ is not reduced to the 0 functional. I corrected this point.

Also, I tried to put together some facts about p < 1 that were dispersed in the article. If I did it right, then the subsection "p < 1" of the section "properties of L^p" could/should be removed. Bdmy (talk) 13:35, 3 November 2008 (UTC)

It should be fine to get rid of my other section. The only thing I think we should be careful about is the reference. Adams theorems are specifically for subsets of Rⁿ. In other comments, we have the sentence

"In this case, the bounded linear functionals on ℓ^p are exactly those that are bounded on ℓ¹,

namely those given by sequences in ℓ^∞." When we say exactly do we mean the dual of ℓ^p is ℓ^∞, and do we have a reference for this? Thenub314 (talk) 15:20, 3 November 2008 (UTC)

Yes, I mean that the dual of ℓ^p is ℓ^∞, and it is essentially the same proof as for ℓ¹; I could say that it is very easy and folklore, but I will try to find a reference (since we have a natural continuous embedding of ℓ^p into ℓ¹, it is clear that what is continuous on ℓ¹ is continuous on ℓ^p; on the other hand, since the standard basis (e_n) is bounded in ℓ^p, the sequence (f(e_n)) of scalars is bounded for every bounded linear functional f on ℓ^p and gives the representation in ℓ^∞). Bdmy (talk) 18:27, 3 November 2008 (UTC)

Sounds good, I will also check to see if I will have a reference in my library. What you say makes sense, but we should still try to find a reference. Thenub314 (talk) 18:36, 3 November 2008 (UTC)

ℓ_n^p is locally convex when p<1

The topology of the space ℓ_n^p is simply the usual topology of Rⁿ, hence it was wrong to say that it is not a locally convex space.

I made some other small changes:

• the description of the duality of ℓ^p is very sketchy. I mentioned that the full story is given later.
  
• precised the statement of Tsirelson's result.
  
• mentioned that the weak L_p-norm is equivalent to a true norm when p > 1, but not when p = 1. I need to find a reference.
  

Bdmy (talk) 20:52, 3 November 2008 (UTC)

Your last bullet is false. This can be seen by taking (1/|x|)^1/p on R. Then the measure of the set where (1/|x|)^1/p>t is the same set as where (1/x)>t^p. Whose measure is 2/t^p. I am not sure about the local convexity for sequence spaces. Keep up the reference search :). Thenub314 (talk) 21:54, 3 November 2008 (UTC)

I don't see how your explanation (that I didn't understand yet) could possibly show that there is no true norm equivalent (hence, with a multiplicative constant) to the weak-L^p functional. Bdmy (talk) 22:19, 3 November 2008 (UTC)

Perhaps better to give you the equivalent norm: assume p > 1, let q be the conjugate exponent and consider

$\sup _{E}\mu (E)^{-1/q}\int _{E}|f|\,d\mu .$

Bdmy (talk) 22:37, 3 November 2008 (UTC)

I misunderstood your claim, I thought you meant weak L^p was equaivalent to L^p. Thenub314 (talk) 08:19, 4 November 2008 (UTC)

Off topic, I know, but are you back from your wikibreak? Thenub314 (talk) 21:54, 3 November 2008 (UTC)

Not exactly. Just don't lay any major projects at my door, and I'll be alright. siℓℓy rabbit (talk) 22:42, 3 November 2008 (UTC)

Hardy spaces for p < 1

At the end of the section on L^p for p < 1, there is this sentence "it is more common to work with the Hardy space H^p whenever possible, as this at least is a Fréchet space: a locally convex complete metric vector space". This seems to indicate that H^p is locally convex in this case. I am 99% sure that this is wrong, for the following reason. Let us work with the unit circle (and analytic functions in the disk). For every ε > 0 and I an interval in the circle, there exists an outer function f_I whose boundary value on the circle has modulus 1 in I and ε elsewhere; taking convex combinations of these functions f_I, for disjoint intervals I of equal measure, we get exactly the phenomenon that makes L^p non locally convex. If nobody tells me why this was supposed to be true, I'll change it sooner or later. Bdmy (talk) 15:59, 8 November 2008 (UTC)

Yes that's right. The Hardy spaces are not locally convex if 0<p<1. siℓℓy rabbit (talk) 16:16, 8 November 2008 (UTC)

I looked it up in Duren's book. The Hahn-Banach theorem fails in the Hardy spaces with p<1, but functionals do separate points. So I have added that to the article. siℓℓy rabbit (talk) 16:29, 8 November 2008 (UTC)

Good; I may add yet another fact later about the "complex convexity" of that Hardy space (something about plurisubharmonic functions).

No, bad idea; I won't add.Bdmy (talk) 10:54, 10 November 2008 (UTC)

Since you seem to watch this article closely, I ask you a question here: the part of the section about properties of \ell_p mentioning Tsirelson is a bit strange in this general article about L^p . Also, one should have more room to explain why people had evidences that \ell_p seems to be everywhere, but isn'it reasonable to copy (not remove) that section as the start of a new article, more with a Banach space flavor? What do you think? Bdmy (talk) 22:22, 8 November 2008 (UTC)

I think a new article is a good idea. There is clearly a need for an article on sequence spaces. I started the article c space to cover c and c₀ some time ago (mostly because I needed an article to point to from the List of Banach spaces). But it is clearly preferable to have all of the little-ell and c spaces dealt with in the same place. I'm not sure how the present article should change to reflect the contents of the new one. siℓℓy rabbit (talk) 12:29, 10 November 2008 (UTC)

OK, but what new title? I had myself problems finding back the article L^p spaces because of the symbols in the title. I can try to spend some time on this new article, perhaps merging your C-space with a new project. Can you suggest a WikiPedia-conformant title? Perhaps what you think is simply "sequence spaces", but that is awfully larger! Bdmy (talk) 12:41, 10 November 2008 (UTC)

the only???

The article says "the space L2 is the only Hilbert space of this class". However, every function in Lp (p>2) is also in L2. then why says the only? Jackzhp (talk) 20:50, 27 December 2008 (UTC)

Well, it means that (for example) the space L⁴([0, 1]), with its own norm, is not a Hilbert space. What problem with the fact that this space is a subset of L²([0, 1])? —Preceding unsigned comment added by Bdmy (talk • contribs) 21:14, 27 December 2008 (UTC)

Lipschitz embedding?

Is it true that the embedding of L^q into L^p, for p < q, on a finite measure space, is Lipschitz? I seem to recall reading that it is (maybe under a further condition?), but I'm not sure where. If it is true, it would be good to include in the article. -- Spireguy (talk) 16:31, 18 August 2009 (UTC)

OK, that's obvious. Somehow got confused on what Hölder gives you. I put in the fact that it's Lipschitz. -- Spireguy (talk) 02:36, 25 August 2009 (UTC)

Section on Embeddings

The fact that on a finite measure space we have the embedding $L^{q}\rightarrow L^{p}$ for $p<q$ follows from the interpolation inequality with suitable choice of exponents, much easier to use than Jensen's since in that context we would also require a probability measure. 91.104.213.243 (talk) 20:45, 13 January 2010 (UTC)

You can regard any finite measure as a probability measure, just divide by the total mass. Hyperbola (talk) 03:14, 12 March 2010 (UTC)

Applications-> Hilbert Spaces

This section states, that all Hilbert-spaces are isomorphic to l^2, but this should only hold for seperable Hilbert Spaces. —Preceding unsigned comment added by 134.76.55.100 (talk) 10:35, 20 January 2010 (UTC)

Merger proposal

It is proposed to merge LP Norm into this article. It's really just a brief definition, which is already found in the bigger article. Only additional information is the term Chebyshev norm, which incidentally redirects to Uniform norm. Favonian (talk) 21:32, 18 March 2010 (UTC)

I'm in favor of merging LP Norm into Lp space. I would appreciate a redirect from LP Norm to the appropriate section of Lp space though. Romanpoet (talk) 00:35, 19 March 2010 (UTC)

There certainly will be such a redirect according to the procedure for merging. Favonian (talk) 12:05, 19 March 2010 (UTC)

From B.T., 03/19 : I disagree. Smaller articles, with concepts kept separate and more easily disgestable help in absorbing these ideas. To a learner like me at least, ever larger and more comprehensive articles can get unwieldy, quite daunting, and so less convenient. Besides, these seem sufficiently distinct notions. (After all, for p < 1, there are Lp-spaces lacking Lp-Norms, aren't there?) —Preceding unsigned comment added by 72.11.103.229 (talk) 23:52, 19 March 2010 (UTC)

From BRM, 2011/01/06: I agree with B.T.. This article is useful to me because is small and easy to make sense of. If I had to fish for this on the larger article I might give up. —Preceding unsigned comment added by 68.101.216.230 (talk) 05:38, 7 January 2011 (UTC)

I've turned LP Norm into a redirect to this article. This seems like the obvious thing to do to me, but if there's really some dispute then I'm sure it can be sorted out. Ozob (talk) 02:41, 7 April 2010 (UTC)

Also, LP Norm is capitalized badly. Maybe it's better off deleted; Lp norm is already a redirect here. Ozob (talk) 02:42, 7 April 2010 (UTC)

Good redirect. There was really nothing to "merge" anyway. Sławomir Biały (talk) 12:20, 8 April 2010 (UTC)

Ummmm....has no one else noticed

The last time I checked, Lp spaces are INFINITE dimensional vector spaces. Moreso, they are normed linear spaces (to use the old but popular terminology) which are complete, i.e. Banach spaces. In the case of L2, it is a Hilbert spaces because its norm is induced by an inner product. The major point though and the reason Lp spaces are the major discussion of Functional Analysis is because they are INFINITE dimensional. —Preceding unsigned comment added by 166.205.10.251 (talk) 02:59, 26 December 2010 (UTC)

169.234.133.52 (talk) 21:14, 5 May 2011 (UTC)The Lp spaces need not be infinite dimensional: If the measure space has finitely many points you obtain finite dimensional Lp spaces.

Question on Lp equivalence classes

According to the article, Lp is defined as a quotient space with two functions f and g identified if f = g almost everywhere. For S = R (or S = [a, b] with some real a < b) and the Lebesgue measure, would each of these equivalence classes also include one Riemann-integrable function? If yes, how can you prove that? If no, is there a counter-example?

In other words: Is it possible for every Lebesgue-integrable function to find a Riemann-integrable function by just changing the function’s value on a set of measure zero?

Background: The standard example of a Lebesgue-integrable function which is not Riemann-integrable is the indicator function of Q or the Cantor Set or some other set of measure zero. Now I am asking myself: Is it possible to construct a real Lebesgue-integrable function which is not just a Riemann-integrable function changed on a set of measure zero? 109.84.0.136 (talk) 20:25, 14 November 2011 (UTC)

Beautiful presentation

You probably should consider using a few pictures from http://www.florianbrucker.de/index.php?p=pnormswarp for a rather beautiful presentation. And beauty matters in math too, as you're well aware. 89.107.83.198 (talk) 03:59, 31 January 2012 (UTC)

L^p,m

Should

(Latex) "L^{p,m}" resp. "L^{p,m}([a,b])", and "L_p^m" resp. "L_p^m([a,b])"

(HTML) "L^p,m" resp. "L^p,m([a,b])", and "L_p^m" resp. "L_p^m([a,b])"

, with real numbers a, b, be mentioned? That later notation (with lower p) is used e.g. in "Introduction to the Theory of Nonlinear Optimization" (p.26) and as far as I see without any explanation. It should be that L^p,m([a,b]) = L^p([a,b]^m). -IP, 16:36, 12 September 2014 (UTC) — Preceding unsigned comment added by 91.63.244.186 (talk)

Undefined jargon

The term "p-norm" in the first sentence is undefined, so the lead section feels incomplete; the concept of Lp space is explained with concepts that are themselves not explained nor reachable from the lead. I see that p-norm is a link to the first section of this same article, so trying to explain the term with a wikilink creates a circular reference, and there's no clear incremental ordering on which concepts should be learn to understand the topic.

I thought of linking "norm" to Norm (mathematics), but I'm not sure that it's the same concept, or that it would be enough explanation. Could someone provide a brief insight in the lead of what a "p-norm" is, so that readers don't have to navigate elsewhere to find it out? Diego (talk) 13:40, 4 December 2014 (UTC)

Misleading "decay" statement in embedding section

Maybe it is just me but I read the statement

"Consider the Lebesgue measure on the half line (0, ∞). A continuous function in L¹ might blow up near 0 but must decay sufficiently fast toward infinity."

as if it implied $f\in C((0,\infty ))\cap L^{1}((0,\infty ))\ \Rightarrow \ \lim _{x\to \infty }f(x)=0$.

However, that claim is not true. The function

$f(x):=\sum _{n\in \mathbb {N} }\left(2n^{4}\left(x-n-{\frac {1}{2}}+{\frac {1}{2n^{3}}}\right)1_{\left[n+{\frac {1}{2}}-{\frac {1}{2n^{3}}},n+{\frac {1}{2}}\right]}(x)+2n^{4}\left(n+{\frac {1}{2}}+{\frac {1}{2n^{3}}}-x\right)1_{\left(n+{\frac {1}{2}},n+{\frac {1}{2}}+{\frac {1}{2n^{3}}}\right)}(x)\right)$

is obviously continuous and triangular on each $\left[n+{\frac {1}{2}}-{\frac {1}{2n^{3}}},n+{\frac {1}{2}}+{\frac {1}{2n^{3}}}\right]$ with $\forall n\in \mathbb {N} :\ f\left(n+{\frac {1}{2}}\right)=n$, i.e., $\lim _{x\to \infty }f(x)$ does not exist.

Furthermore,

$\forall N\in \mathbb {N} :\ \int _{0}^{N}f(x)dx=\sum _{n=1}^{N-1}{\frac {1}{2}}{\frac {1}{n^{3}}}n\leq {\frac {1}{2}}\sum _{n\in \mathbb {N} }{\frac {1}{n^{2}}}<\infty$

shows $f\in L^{1}((0,\infty ))$.

Drownedturtle (talk) 13:49, 14 January 2015 (UTC)

Urysohn's Lemma (Wrong?)

In the section to denseness of continuous functions it is stated the existence of a continuous cutoff.
However the assumption only addresses a metrizable space not necessarily locally compact.
I guess this statement is wrong as for the paris railway metric. --Freeze S (talk) 03:11, 2 March 2015 (UTC)

It also states "More can be said when S is a metrizable topological space and Σ its Borel σ–algebra, i.e., the smallest σ–algebra of subsets of S containing the open sets." before talking about the cutoff functions. Isn't that precisely what you mean, or am I missing something? Drownedturtle (talk) 10:56, 2 March 2015 (UTC)

Upper or Lower-case L

This article sometimes uses upper-case L and sometimes lower-case l. I can't determine if there is supposed to be some pattern to the use or whether they are used interchangeably. It would be good to have an explicit statement about the difference between upper and lower case L. David Poole (talk) 11:35, 28 April 2015 (UTC)

They are not interchangeable. Lower-case l is a special case of upper-case L; namely, lower-case l is an upper-case L of sequences with counting measure. If you look at the beginning of section 2 (The p-norm in countably infinite dimensions and ℓ p spaces), then you will see the comment "For more details on this topic, see Sequence space." Furthermore, it says explicitly what lower-case l spaces are. In section 3 (Lp spaces) it explicitly says what upper-case L spaces are. Drownedturtle (talk) 11:50, 28 April 2015 (UTC)

What is || · ||p ?

Could someone please elaborate on what the symbol || · ||p , specifically the dot instead of a function, in section https://en.wikipedia.org/wiki/Lp_space#Lp_spaces , means. I'm sorry for being dense, but I've been staring at it for well over an hour now, and I imagine I'm not the only one. A one sentence explanation would really help. Please feel free to delete this Talk section if you add that. Zombiejesus (talk) 05:01, 11 June 2015 (UTC)

$\|\cdot \|_{p}$ means the function that maps each $f$ to $\|f\|_{p}$. --Zundark (talk) 10:01, 12 June 2015 (UTC)

"B-norm" etc.

I am removing reference to "B-norm"s here and on the page Norm (mathematics). There is no indication of what the terms mean and the only relevant Google hits I found were these instances in Wikipedia. In this article I removed the parenthetical "(B-norm, with "B" for Banach)". 130.88.123.107 (talk) 13:59, 24 November 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Lp space/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Longer lead and more references would help. Geometry guy 22:02, 17 June 2007 (UTC) The sole definition of 0o=0 may give way to read ¡xi¡p = 0 in Lp space --> Motivation --> When p = 0 For a better comprehension it would be good to remark that when p-->0 ¡xi¡p = 1 if xi =/= 0 —Preceding unsigned comment added by Siniestra (talk • contribs) 04:50, 22 September 2010 (UTC)

Last edited at 04:52, 22 September 2010 (UTC). Substituted at 02:18, 5 May 2016 (UTC)