The Riesz-Thorin Theorem, often called the Riesz-Thorin interpolation theorem or the Riesz-Thorin convexity theorem, is a result about the interpolation of linear operators between Lp spaces, named after Marcel Riesz and his student Olof Thorin. It bounds the norm of a linear map acting between such spaces by using the fact that some Lp spaces, notably the Hilbert space L2 and the simpler spaces L1 and L-infinity, have much simpler structure than the general case, so that a theorem proved in those simple cases can be carried over to the more complicated intermediate cases; the related Marcinkiewicz theorem is similar in spirit but also applies to certain non-linear maps.
Facts
StatementLet (Omega1, Sigma1, mu1) and (Omega2, Sigma2, mu2) be sigma-finite measure spaces, 1 <= p0, q0, p1, q1 <= infinity, and T a linear operator that boundedly maps L^p0(mu1) into L^q0(mu2) and L^p1(mu1) into L^q1(mu2). Then for 0 < theta < 1, T boundedly maps L^ptheta(mu1) into L^qtheta(mu2), with operator norm bounded by the (1 - theta) power of the first norm times the theta power of the second. 1 Partially Attested
Proof YearRiesz stated and proved the interpolation theorem in 1927 with certain restrictions; Thorin's extension to the full range was published in 1938, so the year of the complete Riesz-Thorin theorem differs. Classification
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Source Riesz-Thorin theorem (Wikipedia)
Sources
1. Riesz-Thorin theorem (Wikipedia)
Statement of the theorem
The interpolation theorem was originally stated and proved by Marcel Riesz in 1927.
Historical development, Riesz 1927
originally stated and proved by Marcel Riesz in 1927
Proved By: Marcel Riesz, Lead
The interpolation theorem was originally stated and proved by Marcel Riesz in 1927.
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