Wall socket problem?

I have a wall socket on a wall next to a window sill which has a table lamp connected. This is switched on from a switch in the hall
This has worked well for many years, until recently.

This weekend I replaced the old socket with a new one.
Still the light does not come on.

With an electric tester placed on the terminals at the back of the socket I find that the positive and negative terminal both light up.

Can anyone offer any advice?

Alan.