The idea of using hardening inflatable antennas for nanosatellites is analyzed in this paper. Antennas should be compact in the process of transporting to the orbit and reliable when they are put in to operational state. In addition the antennas should retain its efficiency for a long time. It is proposed to make inflatable antennas from prepreg (textile impregnated with epoxy resin) and carry out a hardening chemical reaction directly in space. To reach this it is efficient to use the hot hardening of the reaction mixture. The prepreg must retain in a soft, easily deformable state for a long time under conditions that are typical for transport and storage of the nanosatellite at the spaceport. Nothing should happen at ordinary temperatures not exceeding 25 degrees Centigrade. The reaction should begin in orbit at high temperatures after inflation of the construction. In this case heating can be done from artificial sources on the nanosatellite. But their energy is undesirable to spend for heating. More attractive is the increasing of the temperature of the prepreg as a result of the action of solar radiation. The aim of the work is to analyze what temperatures will be in the elements of the inflatable antenna and whether additional heating of the antenna is required. It is also important to calculate the warm-up time of the material. Is it substantially less than the time of circulation of the nanosatellite around the Earth which is approximately equal to 1.5 hours? As a result of the computational experiments it was established that hot hardening cannot be carried out without additional heating. However a different result is obtained when there is a thin aluminum film on the surface of the prepreg from the outside of the inflatable element of the antenna. In this case hot hardening is quite feasible.