Krein–Smulian theorem

In mathematics, particularly in functional analysis, the Krein-Smuilian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein and Vitold Shmulyan, who published them in 1940.[1]

Statement

Both of the following theorems are referred to as the Krein-Smulian Theorem.

Krein-Smulian Theorem:[2] Let X be a Banach space and K a weakly compact subset of X (that is, K is compact when X is endowed with the weak topology). Then the closed convex hull of K in X is weakly compact.

Krein-Smulian Theorem:[2] Let X be a Banach space and A a convex subset of the continuous dual space

X^{\prime }

of X. If for all r > 0,

A\cap \left\{x^{\prime }\in X^{\prime }:\left\|x^{\prime }\right\|\leq r\right\}

is weak-* closed in

X^{\prime }

then A is weak-* closed.

References

Krein, M.; Šmulian, V. (1940). "On regularly convex sets in the space conjugate to a Banach space". Annals of Mathematics. Second Series. 41: 556–583. doi:10.2307/1968735. MR 0002009.
Conway 1990, pp. 159-165.

Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)

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[1] Krein, M.; Šmulian, V. (1940). "On regularly convex sets in the space conjugate to a Banach space". Annals of Mathematics. Second Series. 41: 556–583. doi:10.2307/1968735. MR 0002009.

[FOOTNOTEConway1990159-165-2] Conway 1990, pp. 159-165.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Krein–Smulian theorem

Statement

See also

References