Understanding H=L^2 in Quantum Mathematics

[pellman]

Let H=L^2(R^n)

Here H is a Hilbert space, R^n is of course the nth product of the real line. what is L-squared? context is a quantum text for mathematicians.

Later the author uses H=L^2([0,1]) in another example.

 

[nicksauce]

L² is essentially the space of square-integrable functions.
See: http://mathworld.wolfram.com/L2-Space.html

 

[pellman]

That makes perfect sense. Thank you!

 

[daviddoria]

L^2([0,1]) means functions whose square integral from 0 to 1 converges, but what is L^2(R^n)? Functions whose square integral from -\infty to \infty converges? Or any sub interval of R^n?

 

[maze]

It is the integral over all of Rn. For example, if n=3, then \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y,z) dx dy dz, which is also written in more compact form as \int_{\textbf{R}^3} f(\textbf{x}) d\textbf{x}