Given a field , a -vector space pairing between -bialgebras and is a -linear map such that
\langle h\otimes h', \Delta_K k\rangle = \langle h h', k\rangle
\langle \Delta (h), k,\otimes k' \rangle = \langle h, k\otimes k'\rangle
(where on the left hand side denotes the map given by ), is called the bialgebra pairing.
The bialgebra pairing which is perfect as -vector space pairing (i. e. if implies that either or is ) is called the bialgebra duality.
If and are Hopf algebras then the compatibility with antipodes is .