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

jorgdv
Messages
29
Reaction score
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
The space ##L^2(\mathbb R)## is well defined, who told you that it is not?
 
Thank you, I think it was a misunderstanding.
 

Thread '##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?'

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Spherical Harmonics Expansion convergence
Replies
6
Views
2K
A What separates Hilbert space from other spaces?
2
Replies
59
Views
6K
I Quantum particle's state in momentum eigenfunctions basis
Replies
16
Views
2K
I ##L^2## square integrable function Hilbert space
Replies
11
Views
3K
I How to check if a matrix is Hilbert space and unitary?
Replies
16
Views
2K
A The meaning of ##(ict, x_1,x_2,x_3)##
Replies
4
Views
2K
MHB Help with functional analysis questions
Replies
9
Views
2K
I Fourier transforms in Hilbert Space
Replies
2
Views
405
A Hilbert spaces and kets "over" manifolds
Replies
10
Views
3K
A What happens when an operator maps a vector out of the Hilbert space?
2
Replies
59
Views
4K

Hot Threads

Recent Insights

Back
Top