http://it.tinypic.com/r/2iqjvbl/9
Hi guys! I'm writing my thesis for my degree and it's about Sturm-Liouville theory applications. I'm using the book "Al-Gwaiz M.A. Sturm-Liouville Theory and Its Applications" for references and here I have a problem. In the image you can see that I underlined in black the space $L_\rho^2$, where $\rho$ is a function, and its norm because the book does not define it and I don't know/understand its definition. Can someone please help me telling me the definition of this space or some references where I can find it?
It is just a weighted $L^p$-space.
The norm is given by $$ \|u\|_{L^2_\rho} ^p:=\int_D |u(x)|^2 \rho(x) dx, $$ scalar product is $$ (u,v)_\rho = \int_D \rho(x) u(x)v(x)dx. $$