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

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