Lighting a white background is easier than you think.
Lights used in this video:
IntellyTech Light Cannon - https://geni.us/EXGk
ARRI L7-C Fresnel - https://geni.us/qGiWmr
ARRI LoCaster 2 - https://geni.us/Hd5Q0
First, you have to light the background and THEN you have to light the subject separately. You don't need fancy lights, you simply need enough light to cover the dark spots on the background.
Lighting both the subject and background will result in harsh shadows on the background and an overexposed subject.