Dear Robert,

As I put it in my prelude,  a space expands when the "separation" between distinct objects increases as a parameter up on which the space is constructed increases. One of the parameters in the question is the regularity index/exponent and as it increases, the separation between distinct elements of particular differentiablity property, increases. 

@Robert,

the question seem to refer to a situation in which two functions belong to a whole family of Sobolov spaces and that there distance as a function of the index k is under consideration. Of course, the question is not formulated clear enough.

Dear Robert, 

Yes the two spaces are conjoined as  one is an expanded of the other in my construction.  For instance for  Ω=[0,1], the two functions f(x) = cosx and g(x) = sinx w e see 

ρ_{_{W^{1,2}(Ω)}}(cos x, sin x) = √2  ≥ √(1-sin²1) = ρ_{_{ℒ²(Ω)}}(cos x, sin x)

i.e., the two functions f and g are √(1-sin²1) apart in the Hilbert space W^{0,2) where k=0 while they are √2 apart when they are in the Sobolev space W^{1,2} where k = 1. That is what I mean by elements are further apart  when the regularity index k is increasing, which I defined that as expanding. 

Here ρ_{_{W^{1,2}(Ω) and ρ_{_{ℒ²(Ω)}}(cos x,sin x) are distances in the respective spaces. 

I don't like to see modern buzz words like 'dark energy' used for embellishing humble observations. BTW the universe may expand also without dark energy, it is only the specific plan it seems to follow for which dark energy is the common explanation.  

@ Ulrich,

I do not assume that it is a buzz word but an analogy between what a parameter of change does in one structure (accepted in the field) and  what the question refers to in this case. As we in the scientific community know, analogy is one of the strategies to understand and solve problems in science. 

No, the question doesn't make sense. Technical terms have meaning in their context. This isn't the case here. 

@Dejenie

For simplicity let us consider from all your Sobolev-spaces only functions with compact support for which the derivations of arbitrary order exist. The two functions f_1 and f_2 have different distance for different k only by the trivial fact that the number of sum terms used for defining the distance is different. It is hard to understand how one can see such an obviously conventional and technical dependence as one that is connected to aspects of nature. You are right: analogy is a great tool in science. But when used carelessly, it is very efficient in creating blunder.  

@Ulrich,

Is that not what expanding mean in simpler terms, just leaving the coinage I used in my question? Something changes when a parameter(s)  changes. I feel you try to  not like the analogies that are made, after you realized what the actual question was meant. I do not understand your problem of using discrediting words such as "buzz words", "careless". I like to create analogies to create simpler understanding of complex issues. Einstein said, popularly  quoted (not sure if true)  "if you do not make things simpler,  then you do not understand it very well ". For instance I created a thread of a complex financial collapse with the analogy  of a collapse a star to a black hole with hundreds of opinions and several direct appreciation with the analogy I have made.  I have attached the particular thread I am talking about in order you to see it. However let us not digress from the main question I have posted. 

https://www.researchgate.net/post/Is_the_economic_collapse-a_black_hole_syndrome2 

@ Robert,

Yes you have said, as time passes and due to the presence of dark energy, space  expands. Here you included  the other parameter of change which I missed, time. But again time alone does not make the observed expansion but the dark energy. Thus what we observed is the separation between two spatial objects is larger today compared to what that separation was some hundred or thousands of years back. 

The family of metrics you mentioned are over function spaces which are Sobolev spaces. The question is to discuss about these function spaces based on the metrics that are defined in them, keeping all things the same, changing the regularity exponent changes distances between two distinct functions which  I gave an example above. I am not sure why the spaces W^{k,2}(Ω) and W^{k+1,2}(Ω) are not Sobolev spaces but family of metrics. 

@Robert,

Dark energy is not a cosmological constant  but a force that pushes objects apart in the universe, contrary to what gravity does. It is what is believed that causes dark energy, that what a cosmological constant is.  

Dear all,

The question was to show that  for k∈ℕ∪{0} either 

[1]  for f, g ∈ W^((k+1),2)_(Ω) then

      σ_W^((k+1),2)_(Ω)(f,g) ≥ σ_W^(k,2)_(Ω)(f,g)

where σ_(.) is the distance interns of the inner product defined in the respective spaces or simply

[2]  for f ∈ W^((k+1),2)_(Ω)  we have

      ∥f∥_W^((k+1),2)_(Ω) ≥ ∥f∥_W^(k,2)_(Ω).

That is,  length of a function is stretched further in W^((k+1),2)(Ω) from its length in W^((k,2)(Ω) 

Let us make life a fun and keep things simpler. Looking complex things  interns of analogy does give more clarity and more sense than otherwise. 

Albert Einstein said " If we do not explain things simpler, then we do not understand them".  

Dejenie,

your observations [1] and [2] absolutely trivial, not less trivial than Robert's k||  ||-example. Mathematical trivialities don't have the power to make life a fun. Einstein certainly had in mind explanations of non-obvious matters not of trivialities.

@Ulrich,

Take a breath, I can sense  your rage  on your wordings and I am not sure why. All things are trivial when looked from above and complex when looked from bottom.

You reminded me of what I read, a physics journal editor in Germany wrote to Albert Einstein after he received his article for publication. The editor said, Dr. Einstein your work seems more fiction than physics and I regret to inform you that your paper will not be accepted in the physics journal he audited.  He trivialized his works not only to throw his works out, but to disappoint him not to try to send his works to another journal - just memory of events.  

The question is regarding two spaces which have their own distances that are  obtained from canonical inner products that define the spaces. So when we say norm of an element or distance between elements, we mean these canonical norms. The Sobolev space I gave is a well known function space with a known canonical norm that is defined from its inner product and the answer has to be from these perspective.  

Is the word "God particle" too much exaggerated or using your terminology "buzz" word for "Higgs Boson" ? for something that is still illusive and not known definitively  but people use it as a "buzz" word to make the story earth shattering. Science uses "buzz" words to motivate people, to promote ideas , results and make them popular. I am a professor and teacher of mathematics.

I do motivate students by creating analogies in order to make concepts closer to  than far away and illusive from them. When I say mathematics is ubiquitous in nature, every creature does it, as surviving demands making correct or optimal decisions on constrained feasible domain and those who make ill decisions either fail some how or perish. These decisions are mathematical decisions and that means all people do mathematics since child hood but they do not do it through writings, through mathematical expressions and equations we write to solve problems. You will say, these things are just buzz  things.   

@Ulrich,

Just to add what I wrote, I do not claim my question is not trivial or it is complicated, I just asked a question that is trivial or not. What was needed is to give the answer mentioning (if you like) it is trivial. That is it. 

NASA starts Stealing my Gravitation theory:

“The majority of Ferent matter(Dark matter) is the core of the supermassive black hole” Adrian Ferent(Jan 14, 2015)

NASA --"Primordial Black Holes Formed in 1st Fraction of a Second of the Universe Could Function as Dark Matter" (The Weekend Feature)

October 29, 2016

“Black holes formed in the universe's first fraction of a second -- could work as dark matter." Kashlinsky said (May 24, 2016)

http://www.nasa.gov/feature/goddard/2016/nasa-scientist-suggests-possible-link-between-primordial-black-holes-and-dark-matter

http://www.dailygalaxy.com/my_weblog/2016/10/nasa-goddard-primordial-black-holes-formed-in-1st-fraction-of-a-second-of-the-universe-could-functio.html

Ferent matter is Dark matter – Adrian Ferent (Jan 14, 2015)

https://www.youtube.com/watch?v=taXMgfqKgFQ

Dear Dejenie,

please take your sensing my rage as an illusion which is perfectly possible taking the distance between us into account. I would enjoy if you could uncover in a bit of detail the thoughts you found interesting about your finding both from a mathematical and from a physical perspective. As is obvious from my previous answers I was so far not able to find anything here that inspired me. But I'm always open to change my mind.  

The question is trivial, in that it doesn't make sense. That the distance between points in a space depends on a parameter in such a way that it is an increasing function of the parameter doesn't mean that the metric of the space is a solution of Einstein's equations with the parameter as the cosmological constant. 

And as mentioned, the parameter in question labels spaces, that aren't isomorphic. So this makes any discussion even more futile. It doesn't make sense to compare distances between points of spaces that aren't isomorphic.

What"s called dark energy can be described by the cosmological constant. It doesn't require a particular scalar field-which is what's meant by the term ``dark energy'', in contradistinction to the term ``cosmological constant''. 

@Stan,

First trivial does not mean senseless, if you are familiar with mathematics. Second you have to answer the question first and say trivial later (if you want). Why so much fighting without even trying to answer the question, how trivial it may be. There are multitudes of mathematical results which are trivial in that structure. Almost all corollaries are trivial  results of  main results that are called theorems or propositions but they are results to be considered. In fact the beauty of the main theorem or proposition is displayed from the simplicity of the corollary that follows. Besides I do not think you are talking about what the question is and your writings are unrelated to the question any way. You can talk about the length of a function in a particular setting and the length of the same function and its derivative (if the derivative exists or if you consider generalized or distributional derivative) together and there is no need to bring isomorphism here. I gave an example in one of my postings about this.  You changed the topic from a trivial question in mathematics to a sub theory of cosmology or at times it looks a political writing. 

Honesty and  being truthful I think are  pillars of science. Sorry! 

Dear Ulrich,

I do not think distance creates illusion of this kind for otherwise all physicists and cosmologists will be suffering from  immense illusions.  As you asked, is it reasonable to ask you what distance you consider  when you say distance ? the great circle distance of any distance, Euclidean or otherwise and multiplying that by a constant? I think there is a standard distance when you say which is the geographic or great circle distance as we are living on the surface of the spherical globe.

@Robert,

Why W^{{k,2}}(Ω) is a family of metrics while it is perfectly a Sobolev space? Besides as long as the spaces are not identical, of course there are elements in one space that are not contained in the other and therefore will be different. In fact the space with a larger regularity exponent is contained or embedded in the one with a smaller regularity exponent. 

Once more: the question doesn't make sense, that's why it's trivial. It's a syntactically and grammatically correct sentence-but it's meaningless. The set of spaces, labeled by the integer k, doesn't have anything to do with solutions of the Einstein equations with positive cosmological constant,  in the sense asked. Sobolev spaces are used for describing properties of  solutions of partial differential equations, among them Einstein's equations. That's all. It is possible to ask questions that don't make sense. This is one of them.

 @Stan,

Again, trivial is not equivalent to senseless. There is no mentioned of  what you called Einstein equation in my question. I am not sure whether you are talking about my question. 

Dark energy makes sense only in the context of cosmology, that's described by Einstein's equations, that describe gravity, in general, at the level of classical physics. That's why. Once more, that the question is senseless makes it trivial, not the other way around. 

+1 to "this question makes no sense"

Side comment: You don't need to have k a non-negative integer to define Sobolev spaces.

My reading of the question (again, after reading the comments) is that it is saying "should we invent terminology because [I think that] the differentiability index in Sobolev spaces refers to the expansion of the space, which is kind of like an analogy to dark energy in cosmology"

So that is, the question isn't actually trying to say that the actual cosmological constant has something to do with the differentiability of the metric? (If so then it's even more preposterous)

@ Stephen,

I think it is me who creates the analogy 

The fact that it is preposterous gives an attention to it- which is a good coinage of the terms I used in describing the analogy. 

Besides, I did not say the exponents should always be non negative integers. In fact the Slobodeckij spaces are Sobolev spaces of fractional exponents. Traces of functions in Sobolev spaces are functions of fractional exponents. As a matter of fact,the exponents  can be taken throughout the set of all real numbers.  When we consider functionals or conjugate spaces of Sobolev spaces, we have negative exponents as well.  For instance W^{{-1,2}}(Ω) is a space of linear functionals from W^{{-1,2}}(Ω) which is with a negative exponent. W^{{1/2,2}}(∂Ω) is an other such a space with a fractional exponent which is the trace image of the  W^{{1,2}}(Ω).

I would expect you saying, the comments have no meanings than the question. 

What does space expand mean to you and others by the way, although space is  infinitely extended? 

" I did not say the exponents should always be non negative integers"

Yes, you did in the question. I thought it was worth mentioning that there was no point restricting it.

So to clarify, you are just calling "k" the dark energy, motivated by the usual meaning of; you're not actually saying that the differentiability of the metric in general relativity has something to do with the cosmological constant? (This is how I think your question has been read by people here)

Not at all. If that is how they understood the question, then there was a discussion between two aliens of no common language or as what happened in Babylonia in religious texts, in which the people of Babylon dispersed without achieving what they  intend to build, a tower that will enable them to reach to the heaven, because of  a sudden lose of a common language to communicate. 

 Dear all,

We know that W^{(k+1,2)}(Ω) ↪ W^{(k,2)}(Ω), where the symbol ↪ indicates embedding or containment.  Thus

∀f∈W^{(k+1,2)}(Ω) ⇒ f∈W^{(k,2)}(Ω)

But ∥f∥_{W^{(k+1,2)}(Ω)} ≥ ∥f∥_{W^{(k,2)}(Ω)}. What is the cause the inequality for the function indicated?

I think I'm going to regret coming back to this thread, but here goes:

I assume first the inequality is meant to be a W^{k,2} norm on the right hand side? In which case, the inequality is because by definition (depending exactly on which norm you use, since there are equivalent ones that might introduce a constant) the left hand side is equal to the right hand side plus the L^2 norm of the (k+1)th weak derivative. And then this inequality implies the inclusion of one Sobolev space in the other.

Oh Dejenie,

your last k+1 is certainly a mistake and should be replaced by k. The answer to your question is that  adding positive numbers to a positive number a  increases a. Deep wisdom ?

Haha Ulrich! Yes, that's the more direct answer :) 

Ulrich,

Thank you for your pointer. It was a typing error. 

Stephen,

What is the canonical norm of the space W^{(k+1,2)}(Ω) ?

There are equivalent norms that induce the same topology, but the usual one I'd use is just given by the sum of the L^2 norms of the first k+1 weak derivatives (including the zeroth, the function itself).

This really has absolutely no connection to dark energy though. Of course you are permitted to make up your own terminology for the differentiability index, but it means nothing to anybody else

I did not ask the existence of equivalent norms and the question is not that. When someone asks you what makes your weight on earth and Jupiter vary, it is a reasonable question which has reasonable answer but your are talking about equivalent  weights on each planet you might have.  You do not have to simply go out and say unrelated things that  have nothing to do with the question. The space I consider is an inner product space and the norm I am considering is the canonical norm  that is generated from that inner product.  I am comparing such norms on the two spaces with all parameters remain the same including the domains of definition except one parameter and finding reason(s) what causes that inequality is my question. You can say whatever you want and  I will say what I think is reasonable, reasonable to make analogies, even if they have very minimal meanings.