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

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SUMMARY

The discussion clarifies that the space L²(ℝ) is indeed well-defined, contrary to the initial query suggesting it might not be. The participant raised a question about the bounded limits of L²[a,b] and the possibility of defining L²(ℝ) with unbounded limits. It was confirmed that the sum of square integrable functions remains square integrable, and all properties of Hilbert spaces apply. Therefore, L²(ℝ) is a valid space for functions where the integral of the square is finite over the entire real line.

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  • Understanding of L² spaces in functional analysis
  • Familiarity with Hilbert space properties
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  • Basic concepts of square integrable functions
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jorgdv
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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.
 

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