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A topological space is a loop space if it has a delooping. It is an infinite loop space if this delooping has itself a delooping, and so on.
In homotopy theory infinite loop spaces with a choice of deloopings are equivalent to connective spectra.
Infinite loop spaces with a choice of deloopings are the grouplike E-∞ algebras in Top (grouplike E-∞ spaces).
See for instance (Adams, pretheorem 2.3.2) and the references listed there for traditional accounts. See (Lurie, section 5.1.3) for a modern formulation.
(Compare to how just loop spaces are the grouplike A-∞ algebras, see looping and delooping.)
See at free infinite loop space.
Ib Madsen, Victor Snaith, Jørgen Tornehave, Infinite loop maps in geometric topology, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 81, Issue 3, (1977)(doi:10.1017/S0305004100053482)
Peter May, Infinite loop space theory, Bull. Amer. Math. Soc. 83 4 (1977) 456-494 [pdf, doi:10.1090/S0002-9904-1977-14318-8]
Infinite loop space theory revisited [pdf]
John Adams, Infinite loop spaces, Hermann Weyl lectures at IAS, Annals of Mathematics Studies 90 Princeton University Press (1978) [ISBN:9780691082066, doi:10.1515/9781400821259]
Peter May, The uniqueness of infinite loop space machines, Topology, vol 17, pp. 205-224 (1978) (pdf)
Theorem 5.2.6.15 of
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