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

[jorgdv]

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

 

[Hawkeye18]

The space $L^2(\mathbb R)$ is well defined, who told you that it is not?

 

[jorgdv]

Thank you, I think it was a misunderstanding.