The model of population protocols refers to the growing in popularity theoretical framework suitable for studying pairwise interactions within a large collection of simple indistinguishable entities, frequently called agents. In this paper the emphasis is on the space complexity in fast leader election via population protocols governed by the random scheduler, which uniformly at random selects pairwise interactions from the population of n agents. The main result of this paper is a new fast and space optimal leader election protocol. The new protocol operates in parallel time O(log2 n) equivalent to O(n log2 n) sequential pairwise interactions, in which each agent utilises O(log log n) states. This double logarithmic space utilisation matches asymptotically the lower bound 1 2 log log n on the number of states utilised by agents in any leader election algorithm with the running time o( n polylog n), see . Our solution relies on the concept of phase clocks, a fundamental synchronisation and coordination tool in the field of Distributed Computing. We propose a new fast and robust population protocol for initialisation of phase clocks to be run simultaneously in multiple modes and intertwined with the leader election process. We also provide the reader with the relevant formal argumentation indicating that our solution is always correct and fast with high probability.