Per Hjalmar Enflo (born May 20, 1944) is a Swedish mathematician and concert pianist. He is known for his significant contributions to functional analysis, particularly in Banach spaces.
Biography Enflo was born in Stockholm, Sweden. He pursued his higher education at Stockholm University, where he earned his Ph.D. in 1970 under the supervision of Gert Eneström.
Mathematical Contributions Enflo is renowned for solving several long-standing problems in functional analysis:
- The Basis Problem: In 1973, Enflo solved the basis problem, demonstrating that there exists a separable Banach space without a Schauder basis. This disproved a conjecture by Stanisław Mazur from the Lwów Scottish Café Book.
- The Invariant Subspace Problem: He provided a counterexample to the invariant subspace problem for Banach spaces, showing that there exists a separable Banach space with a bounded linear operator that has no non-trivial closed invariant subspace. This result was particularly influential, though a counterexample for Hilbert spaces remains an open problem.
- The Compact Approximation Property: Enflo also showed that there exist Banach spaces that do not have the compact approximation property.
- Uniformly Smooth and Uniformly Convex Banach Spaces: He made significant contributions to the theory of uniformly smooth and uniformly convex Banach spaces, proving the uniform non-squareness of these spaces.
His work often involved constructing highly intricate counterexamples, which required significant ingenuity and deep understanding of the properties of Banach spaces. His proofs were often very long and complex.
Concert Pianist In addition to his mathematical career, Per Enflo is an accomplished concert pianist. He has performed extensively, often combining his tours with academic visits. His repertoire includes classical works by composers such as Chopin, Liszt, and Beethoven. He has frequently incorporated mathematical themes or insights into his musical performances and lectures.
Academic Career Enflo has held academic positions at various institutions, including the Institute for Advanced Study in Princeton, Stanford University, and Kent State University, where he is currently a professor emeritus.