Question about vector space l^p

In summary, the conversation discusses the conditions for {1/n^x} to be an element of l^p where x>0 and 0<p<=infinity. It is established that if x=1, {1/n^x} is an element of l^p for p>=2, and if x<1, then only for p=infinity will it be an element of l^p. The concept of the l^p norm is also mentioned in relation to the series 1/n^x, with the conclusion that it will converge for p>x. The conversation ends with a discussion on the containment of l^p in l^q, with the conclusion that it is true for all finite q and for p=in
  • #1
Ed Quanta
297
0
So if we let x>0, For which 0<p<=infinity is {1/n^x} an element of l^p?

If x=1, then 1/n^x is clearly an element of l^p for p>=2, since for all these vector spaces, the series of 1/n will converge?

But if x<1, then in it seems that only for p=infinity, will {1/n^x} be an element of l^p. Is this correct?
 
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  • #2
Why don't you work out the length of 1/n^x in the l_p norm.
 
  • #3
The series 1 +1/2+1/3+...+1/n doesn't converge. Now if we have the summation of 1/n^x, and x>1, will 1 +1/2^x + 1/3^x + ... + 1/n^x converge? If this does always converge, then as long as p>x the series of {1/n^x} will be an element of l^p,right? That is what I meant to ask before.
 
  • #4
the sum of 1/n^r converges if and only if r>1. HOpefully you can use that with the l^p norm: what is the l^p norm of the series 1/n^x?
 
  • #5
[ (1/1)^p + (1/2^x)^p +...+ (1/n^x)^p]^(1/p)

And as I stated in my previous post, this should only converge where p>x unless I am missing something.


One more question. If p<q, then why must l^p be a subset of l^q? I know that there are two cases to prove here. One in which q is finite, and one in which q is infinity. But I am not sure how I can show that this is true.
 
  • #6
If you think all the series are finite, then yes, you are missing something.

I think you missing something else too.

(a_1,a_2,...) is in l^p iff the sum of (a_i)^p converges, right?

so a_r = 1/r^x is in l^p iff the sum of (1/r^x)^p converges, which is iff px>1.

Right? Or are we talking at cross purposes? It should be clear now why the containment you mention is true.
 
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  • #7
It isn't clear to me where p = infinity
 
  • #8
What isn't clear to you? (a_1,a_2,...) is in L^{infinity} if and only if its terms are bounded, so 1/n^x is in it iff x>0 (which agrees with the notion of px>1 as it happens, as p tends to infinity.

Or do you mean the containment?

If a sum converges its terms tend to zero, and in particular are bounded so its l infinity.
 

1. What is a vector space l^p?

A vector space l^p is a mathematical concept that represents a collection of vectors that can be added together and multiplied by a scalar to produce another vector within the same space. The l^p notation refers to the sum of the absolute values of the vector components raised to the power p.

2. What are the properties of a vector space l^p?

A vector space l^p must satisfy the following properties: closure under vector addition, closure under scalar multiplication, associativity of addition, commutativity of addition, existence of an additive identity, existence of an additive inverse, associativity of scalar multiplication, distributivity of scalar multiplication over vector addition, and distributivity of scalar multiplication over scalar addition.

3. What is the difference between l^1 and l^2 vector spaces?

The main difference between l^1 and l^2 vector spaces is the way in which the vectors are measured. L^1 measures the sum of the absolute values of the vector components, while l^2 measures the square root of the sum of the squares of the vector components. This leads to differences in how the two spaces are structured and their properties.

4. Can a vector space l^p have an infinite number of dimensions?

Yes, a vector space l^p can have an infinite number of dimensions. This is because the dimensions of a vector space are determined by the number of independent vectors needed to span the space, and in an infinite-dimensional space, there is no limit to the number of independent vectors that can be included.

5. How is a vector space l^p used in real-world applications?

Vector spaces l^p have a wide range of applications in fields such as physics, engineering, economics, and computer science. They are used to model physical systems, analyze data, and solve optimization problems. For example, l^2 vector spaces are commonly used in signal processing and image compression algorithms, while l^1 vector spaces are used in the analysis of financial portfolios and in machine learning algorithms.

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