dual of a quotient

Quotients and subspaces trade places under dualization, and the dual of a quotient is the cleanest instance. The quotient V/U collapses a subspace U to zero; a functional on V/U is therefore exactly a functional on V that did not see U in the first place — that is, one vanishing on U. So (V/U)* is naturally identified with the annihilator U^0.

The reason is the universal property of the quotient. A functional g on V/U corresponds to a functional f on V satisfying f(u) = 0 for all u in U (f factors through the quotient). Matching g to that f is a bijection that respects addition and scaling, so it is an isomorphism (V/U)* = U^0. No basis is needed; it is canonical.

There is a dual partner for subspaces: the dual of a subspace U is V*/U^0, that is U* = V*/U^0. Restricting a functional to U is onto, and its kernel is exactly U^0, so the first isomorphism theorem hands you U* = V* / U^0. Duality therefore swaps 'sub' and 'quotient': the dual of a quotient is a sub(space of the dual), and the dual of a sub is a quotient.

Counting confirms the picture: dim(V/U) = dim V - dim U and dim U^0 = dim V - dim U agree, as they must for an isomorphism. This sub-quotient swap is one of the most useful structural reflexes in the subject — whenever a construction quotients, expect its dual to restrict to a subspace, and vice versa.