Bochner

Bochner is a surname associated with several concepts in mathematics, most prominently tied to Salomon Bochner, a Czech-born American mathematician (1899–1986). His work spanned harmonic analysis, probability, and differential geometry, and several foundational results bear his name.

In analysis, Bochner introduced the Bochner integral, a definition of integration for functions taking values in a Banach space, generalizing the Lebesgue integral and underpinning modern vector-valued integration. He also formulated Bochner's theorem, characterizing which functions on a locally compact abelian group are the Fourier transforms of finite measures on the dual group; this theorem links positive-definite functions with measures and is central in probability and harmonic analysis.

In geometry, Bochner is known for Bochner's formula (Bochner identity), an equation relating the Laplacian of the squared norm of a vector field to its covariant derivative and the Ricci curvature, used in vanishing theorems and curvature studies.

The term Bochner space refers to Bochner L^p spaces, spaces of Bochner-integrable functions from a measure space into a Banach space, widely used in functional analysis and applications.

Bochner’s contributions helped unify ideas across analysis, probability, and geometry, and his work continues to influence both theoretical developments and applications in these fields.