# Dunford–Schwartz theorem

Dunford–Schwartz theorem In mathematics, particularly functional analysis, the Dunford–Schwartz theorem, named after Nelson Dunford and Jacob T. Schwartz, states that the averages of powers of certain norm-bounded operators on L1 converge in a suitable sense.[1] Statement of the theorem {displaystyle {text{Let }}T{text{ be a linear operator from }}L^{1}{text{ to }}L^{1}{text{ with }}|T|_{1}leq 1{text{ and }}|T|_{infty }leq 1{text{. Then}}} {displaystyle lim _{nrightarrow infty }{frac {1}{n}}sum _{k=0}^{n-1}T^{k}f} {displaystyle {text{exists almost everywhere for all }}fin L^{1}{text{.}}} The statement is no longer true when the boundedness condition is relaxed to even {displaystyle |T|_{infty }leq 1+varepsilon } .[2] Notes ^ Dunford, Nelson; Schwartz, J. T. (1956), "Convergence almost everywhere of operator averages", Journal of Rational Mechanics and Analysis, 5: 129–178, MR 0077090. ^ Friedman, N. (1966), "On the Dunford–Schwartz theorem", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 5 (3): 226–231, doi:10.1007/BF00533059, MR 0220900. hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

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