Dunford–Schwartz theorem

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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[edit]

\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}


 \text{exists almost everywhere for all }f\in L^1\text{.}

The statement is no longer true when the boundedness condition is relaxed to even \|T\|_\infty\le 1+\varepsilon.[2]


  1. ^ Dunford, Nelson; Schwartz, J. T. (1956), "Convergence almost everywhere of operator averages", J. Rational Mech. Anal. 5: 129–178, MR 77090 .
  2. ^ Friedman, N. (1966), "On the Dunford–Schwartz theorem", Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 (3): 226–231, doi:10.1007/BF00533059, MR 220900 .