Suppose $w(t)$ is a positive and measurable function on $[a,b]$. If $x(t)$ is a measurable real function on $[a,b]$ satisfying $$ \left\| x \right\|^2 =\int\limits_{\left[ a,b \right]}{w\left( t \right) \left| x\left( t\right) \right|}^2dt<+\infty $$ then we say $x(t)$ is a square integrable function with respect to the weight function $w(t)$. We use $L^2\big([a,b];w(t)\big) $ to represent the set of all square integrable function with respect to the weight function $w(t)$.
I want to show that $L^2\big([a,b];w(t)\big)$ is a Hilbert space: I have already proved it is a normed space, but I don't know how to prove its completeness and separability.
I know how to prove the completeness and separability of $L^2([a,b])$, and I wonder if there is some inequality like the (may be wrong) following one $$ \int\limits_{\left[ a,b \right]}{\left| x_n\left( t \right) -x_m\left( t \right) \right|}^2dt<\int\limits_{\left[ a,b \right]}{w\left( t \right) dt}\int\limits_{\left[ a,b \right]}{\left| x_n\left( t \right) -x_m\left( t \right) \right|}^2dt $$ which allows me to use the completeness and separability of $L^2([a,b])$.
Thank you for sharing your ideas.