Why there's no L^2[-inf,inf] space?

In summary, the conversation discusses the space ##L^2[a,b]## and its requirement of bounded limits. The question arises as to why the space cannot be defined for unbounded functions with a condition of ##\int_{-\infty}^\infty |f(x)|^2 dx \le M## for some positive ##M##. The response clarifies that the space ##L^2(\mathbb R)## is indeed well defined.
  • #1
jorgdv
29
0
Hello,

Maybe it's a silly question, but why the space ##L^2[a,b]## has always to have bounded limits? Why can't we define the space of functions ##f(x)## where ##x \in \mathbb{R}## and ##\int_{-\infty}^\infty |f(x)|^2 dx \le M## for some ## M \in \mathbb{R^+}##? As far as I know the sum of two square integrable functions is still a square integrable function, even if it's domain is not bounded; and the rest of properties of Hilbert spaces hold. So why it's not well defined?

Thank you
 
Physics news on Phys.org
  • #2
The space ##L^2(\mathbb R)## is well defined, who told you that it is not?
 
  • #3
Thank you, I think it was a misunderstanding.
 

Related to Why there's no L^2[-inf,inf] space?

1. Why isn't there an L^2[-inf,inf] space?

The L^2[-inf,inf] space, also known as the square integrable space, does not exist because it is not a complete metric space. This means that it does not contain all of its limit points, making it incomplete.

2. What is a complete metric space?

A complete metric space is a space where all Cauchy sequences converge to a point within the space. In other words, it contains all of its limit points and is considered "closed" under limits.

3. Is there a similar space to L^2[-inf,inf] that does exist?

Yes, there is a similar space to L^2[-inf,inf] that does exist, called the L^2[0,1] space. This space contains all square integrable functions on the interval [0,1] and is a complete metric space.

4. What is the significance of the L^2[-inf,inf] space?

The L^2[-inf,inf] space is important in mathematical analysis and functional analysis, as it is used to study various properties of functions and operators. It also has applications in physics, particularly in quantum mechanics.

5. Are there any other spaces that are incomplete like L^2[-inf,inf]?

Yes, there are many other spaces that are incomplete, such as L^p spaces for p < 1 and various function spaces. Incomplete spaces can still be useful in mathematical analysis, but they may require additional tools and techniques to study their properties.

Similar threads

I Spherical Harmonics Expansion convergence
    • Linear and Abstract Algebra
Replies
6
Views
2K
I Quantum particle's state in momentum eigenfunctions basis
    • Quantum Physics
Replies
16
Views
404
I ##L^2## square integrable function Hilbert space
    • Calculus
Replies
11
Views
324
A What separates Hilbert space from other spaces?
    • Linear and Abstract Algebra
2
Replies
59
Views
5K
A Hilbert spaces and kets "over" manifolds
    • Differential Geometry
Replies
10
Views
2K
I How to check if a matrix is Hilbert space and unitary?
    • Linear and Abstract Algebra
Replies
16
Views
2K
I Representation of Spin 1/2 quantum state
    • Quantum Physics
2
Replies
61
Views
1K
MHB Help with functional analysis questions
    • Linear and Abstract Algebra
Replies
9
Views
2K
I Why is this function not ##L^1(\mathbb{R} \times \mathbb{R})##?
    • Topology and Analysis
Replies
6
Views
917
I Question regarding a Free particle and Hilbert space (QM)
    • Quantum Physics
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top