On some properties of weighted Hilbert spaces

We describe the weighted Hilbert spaces $ L_{2,w}(\Omega) $ with positive weight functions $ w(x) $ which are summable on every bounded interval. We give sufficient condition for $ L_{2,w_1}(\Omega) $ space to be extension of $ L_{2,w_2}(\Omega) $ space. We also describe how to use given result in statistical probability density estimation.