L^p spaces are not equivalent for infinite measure sets

In summary, the conversation discusses finding a function f that is in the L^p(R) space but not in the L^q(R) space, where p and q are different. The participants suggest using a function that only blows up when raised to the qth power and suggest using a compactly-supported function, with the example of f = sin being ruled out as it is not in any L^p(R) space.
  • #1
JasonJo
429
2
Find a function f such that f is in L^P(R) but not in L^Q(R) for p not equal to q, where R is the set of real numers.

I'm guessing I need to find a function that only blows up when it is raised to the qth power, but I am having some difficulty proving this.
 
Physics news on Phys.org
  • #2
Try finding some function f that's in L^1 but not in L^2. To make things easier, you can choose a compactly-supported f.
 
  • #3
I suppose just f = sin would do!
 
  • #4
It wouldn't, because sin isn't in any L^p(R) space. :-p
 
  • #5
Oh, right!:blushing:
 

FAQ: L^p spaces are not equivalent for infinite measure sets

What are L^p spaces and why are they important in mathematics?

L^p spaces are a type of functional space in mathematics that are used to study the properties of functions, particularly in the field of analysis. They are defined as the set of all functions that are pth power integrable over a given measure space. These spaces are important because they allow for a rigorous and structured way to study functions and their properties, and they have applications in many areas of mathematics, such as probability, statistics, and differential equations.

What does it mean for L^p spaces to be equivalent?

In mathematics, two spaces are considered equivalent if they have the same structure and properties, even if they may look different on the surface. For L^p spaces, this means that they have the same mathematical structure and behave in the same way, even if they may have different formulas or representations.

Why are L^p spaces not equivalent for infinite measure sets?

This is because the measure of an infinite set is undefined or infinite, which leads to different behaviors and properties for functions in L^p spaces. For example, the integral of a function over an infinite set may not converge, which changes the way the space behaves.

What are some consequences of L^p spaces not being equivalent for infinite measure sets?

One consequence is that certain theorems or properties that hold for finite measure sets may not hold for infinite measure sets. This can make it more challenging to study functions on these spaces and may require different approaches or techniques. Additionally, it highlights the importance of specifying the measure when working with L^p spaces.

Are there any practical applications of this concept?

Yes, there are many practical applications of this concept in various fields of mathematics and science. For example, in probability theory, the use of L^p spaces with infinite measure sets allows for the study of stochastic processes, which are random processes that evolve over time. In physics, L^p spaces are used in quantum mechanics to study the behavior of wave functions. In engineering, they are used in signal processing and image analysis to study and manipulate signals and images.

Similar threads

Difficulty proving a relation is an equivalence relation
Replies
3
Views
2K
Prove every subset of countable set is either finite or else countable
Replies
1
Views
1K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
Replies
3
Views
1K
I Quantum particle's state in momentum eigenfunctions basis
Replies
16
Views
1K
I What is the magnetic equivalent to the Coulomb?
Replies
6
Views
623
Inverse image of Lebesgue set and Borel set
Replies
8
Views
2K
Proving Completeness property of ##\mathbb{R}## using Dedekind cuts
Replies
20
Views
3K
Composition of a Continuous and Measurable Function
Replies
2
Views
3K
Logic: proof that [p -> (q ^ r)] <=> [(p ^ -q) -> r]
Replies
3
Views
2K
How Do Vector Spaces of Linear Maps Differ from Standard Vector Spaces?
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top