Simply put, no. A 400 watt HPS will generate a lot of heat, and also, plants need fresh air or C02 augmentation (properly done is very expensive) to thrive in any circumstances. How would you control the temperature?
IMO Flo-thru fresh air ventilation is the most effective/cheapest. Without fresh air or at least exchanged air of some sort, I doubt that your plants would even survive.
It would be a good idea to cut a couple holes in the closet door or something to allow for real air flow.
My last and final piece of advice would be to read a grow guide. Good luck.