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

Characterizations

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Let $X$ be a Banach space. Then the following conditions are equivalent:

$X$ is a Grothendieck space,
for every separable Banach space $Y,$ every bounded linear operator from $X$ to $Y$ is weakly compact, that is, the image of a bounded subset of $X$ is a weakly compact subset of $Y.$
for every weakly compactly generated Banach space $Y,$ every bounded linear operator from $X$ to $Y$ is weakly compact.
every weak*-continuous function on the dual $X^{\prime }$ is weakly Riemann integrable.

Examples

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Every reflexive Banach space is a Grothendieck space. Conversely, it is a consequence of the Eberlein–Šmulian theorem that a separable Grothendieck space $X$ must be reflexive, since the identity from $X\to X$ is weakly compact in this case.
Grothendieck spaces which are not reflexive include the space $C(K)$ of all continuous functions on a Stonean compact space $K,$ and the space $L^{\infty }(\mu )$ for a positive measure $\mu$ (a Stonean compact space is a Hausdorff compact space in which the closure of every open set is open).
Jean Bourgain proved that the space $H^{\infty }$ of bounded holomorphic functions on the disk is a Grothendieck space.^[1]

References

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^ J. Bourgain, $H^{\infty }$ is a Grothendieck space, Studia Math., 75 (1983), 193–216.

J. Diestel, Geometry of Banach spaces, Selected Topics, Springer, 1975.
J. Diestel, J. J. Uhl: Vector measures. Providence, R.I.: American Mathematical Society, 1977. ISBN 978-0-8218-1515-1.
Shaw, S.-Y. (2001) [1994], "Grothendieck space", Encyclopedia of Mathematics, EMS Press
Khurana, Surjit Singh (1991). "Grothendieck spaces, II". Journal of Mathematical Analysis and Applications. 159 (1). Elsevier BV: 202–207. doi:10.1016/0022-247x(91)90230-w. ISSN 0022-247X.
Nisar A. Lone, on weak Riemann integrability of weak* - continuous functions. Mediterranean journal of Mathematics, 2017.

Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Functional analysis (topics – glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic/Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category

Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category

source: https://en.wikipedia.org/wiki/Grothendieck_space

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