# Algebraic topology – Class space loop space

Leave $$U (n)$$ be the unit group $$n$$-matrices My question is why we have the following isomorphism.

$$Omega BU (n) simeq U (n)$$

where $$BU (n) = E / U (n)$$ is the classifying space and $$Omega BU (n) = Hom (S ^ 1, BU (n))$$ denotes the space of the loop.

Is $$Omega BU (n) simeq U (n)$$ above also a $$H$$-equivalence (therefore iso of $$H$$-spaces)?