A space heater requires a considerable amount of power (most small units start at 500-1000 watts). A small system like this would not have nearly enough capacity. From the hugely inadequate solar panel rating of sub-two watts, it is evident that such a heater could not be run from solar power alone, but you'd have a hard time even running it from the battery. A 500-watt, 120V space heater draws around 4 amps from the wall. If you want to run the same heater from an inverter, the wattage will be the same, but since the voltage is reduced by an order of magnitude, you will now need 40 amps from the battery. Most batteries' amp-hour rating is based on a so-called 20-hour rate, or how much capacity the battery has when discharged over 20 hours. For this battery at 7.5Ah, the 20-hour current rating is about 400mA. At 100 times this load, expect battery capacity to drop by a couple to several Ah, which cuts your runtime - if it is able to run the load at all - to ten minutes or less. So in summation: no, this system would not run a space heater. It's much better suited to small loads. To the author, nice packaging, though.