Grothendieck space
In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space X {\displaystyle X} in which every sequence in its continuous dual space X ′ {\displaystyle X^{\prime }} that converges in the weak-* topology σ ( X ′ , X ) {\displaystyle \sigma \left(X^{\prime },X\right)} (also known as the topology of pointwise convergence) will also converge when X ′ {\displaystyle X^{\prime }} is endowed with σ ( X ′ , X ′ ′ ) , {\displaystyle \sigma \left(X^{\prime },X^{\prime \prime }\right),} which is the weak topology induced on X ′ {\displaystyle X^{\prime }} 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.