EBERLEIN �ŠMULIAN THEOREM

In the mathematical field of functional analysis, the Eberlein–Šmulian theorem (named after William Frederick Eberlein and Witold Lwowitsch Schmulian) is a result that relates three different kinds of weak compactness in a Banach space.

Statement

Eberlein–Šmulian theorem: ^[1] If X is a Banach space and A is a subset of X, then the following statements are equivalent:

each sequence of elements of A has a subsequence that is weakly convergent in X
each sequence of elements of A has a weak cluster point in X
the weak closure of A is weakly compact.

A set A (in any topological space) can be compact in three different ways:

Sequential compactness: Every sequence from A has a convergent subsequence whose limit is in A.
Limit point compactness: Every infinite subset of A has a limit point in A.
Compactness (or Heine- Borel compactness): Every open cover of A admits a finite subcover.

The Eberlein–Šmulian theorem states that the three are equivalent on a weak topology of a Banach space. While this equivalence is true in general for a metric space, the weak topology is not metrizable in infinite dimensional vector spaces, and so the Eberlein–Šmulian theorem is needed.

Applications

The Eberlein–Šmulian theorem is important in the theory of PDEs, and particularly in Sobolev spaces. Many Sobolev spaces are reflexive Banach spaces and therefore bounded subsets are weakly precompact by Alaoglu's theorem. Thus the theorem implies that bounded subsets are weakly sequentially precompact, and therefore from every bounded sequence of elements of that space it is possible to extract a subsequence which is weakly converging in the space. Since many PDEs only have solutions in the weak sense, this theorem is an important step in deciding which spaces of weak solutions to use in solving a PDE.

References

^ Conway 1990, p. 163.

Bibliography

Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
Diestel, Joseph (1984), Sequences and series in Banach spaces, Springer-Verlag, ISBN 0-387-90859-5.
Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
Whitley, R.J. (1967), "An elementary proof of the Eberlein-Smulian theorem", Mathematische Annalen, 172 (2): 116–118, doi: 10.1007/BF01350091, S2CID 123175660.

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[FOOTNOTEConway1990163-1] Conway 1990, p. 163.

[1]

v t e 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

v t e 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

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