The Riesz–Thorin theorem is a fundamental result in functional analysis, specifically in the study of interpolation of linear operators between L^p spaces. It provides a powerful method for establishing the boundedness of a linear operator that is bounded on two different L^p spaces, allowing us to extend this boundedness to intermediate spaces.
Articles by others on the same topic
For a linear operator bounded from to with norms , , interpolate reciprocals by and . The intermediate operator norm is at most . The complex interpolation proof first treats simple functions; completeness and continuity on the endpoint sum spaces identify the extension with the original operator. On finite counting spaces it also gives norm bounds between sequence spaces.