Glove compartment light in 2003 Accord
 

My daughter just mentioned to me while driving with my headlights on,
that the glove compartment light is on even with its door shut. I
asked her to reshut it but same result. When I turned off my
headlights, it went off. I would think the compartment's bulb would
burn out sooner than normal but to be honest I never noticed it before
and for all I know it could have been that way since day one.

Is this normal?