In this paper, a new transmission scheme, signal space cognitive cooperation, is introduced by applying the idea of signal space diversity in an underlay spectrum sharing decode-and-forward multirelay cooperative network. In the proposed structure, the secondary source signal is rotated by a certain angle and then the source and the secondary best relay transmit the in-phase and the quadrature components of the rotated signal. As a consequence, two source signals, rather than one, are transmitted in two-time slots which improves data rates considerably, compared to the conventional cognitive cooperative schemes. In this paper, proactive relaying mode is used in which the best relay is selected based on the max-min selection criterion before executing the transmission. Considering both statistical and the instantaneous channel state information of the feedback channel between the primary receiver and the secondary network, two power allocation methods are adopted at the source. For both methods, closed-form expressions of error probability are derived. Moreover, asymptotic analysis is performed and diversity gain is obtained to provide further insights about the system performance. Finally, analytical expressions are verified by Monte Carlo simulations.