In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space in which every sequence in its continuous dual space that converges in the weak-* topology (also known as the topology of pointwise convergence) will also converge when is endowed with which is the weak topology induced on by its bidual. Said differently, a Grothendieck space is a Banach space for which a sequence in its dual space converges weak-* if and only if it converges weakly.
Let be a Banach space. Then the following conditions are equivalent: