There is a motivation for studying harmonic 2-spheres in a loop space ?G,where gauge group G is a compact Lie group,namely,there is a conjecture,that a parameter space of based harmonic maps into loop space of a degree k is in a bijective correspondence with a parameter space of k-Yang-Mills connections over S4 with a group G modulo based gauge transformations.A proven fact is the Atiyah-Ward theorem which states the bijective correspondence between a parameter space of based holomorphic 2-spheres in a loop space of a degree k and a space of k-instantons over S4 with a gauge group G modulo based gauge transformations.We study harmonic 2-spheres in a loop space due to loop spaces emdedding into infinite dimensional analogue of Grassmann manifold and twistor approach to such maps.