**Background** - **German Mathematician:** David Hilbert (1862-1943) was a prominent German mathematician with a wide-ranging impact on the field. - **University of Göttingen:** He spent most of his career at the University of Göttingen, a center of mathematical research at the time. - **Broad Influence:** Hilbert made significant contributions to algebra, geometry, logic, foundations of mathematics, and theoretical physics. **Hilbert Space: Origins and Impact** - **Not a Single Invention:** Hilbert didn't invent Hilbert space in the sense of sitting down and defining it from scratch. The concept evolved gradually from his work on integral equations and infinite-dimensional spaces. - **Formalization:** Hilbert played a crucial role in formalizing and axiomatizing the notion of Hilbert spaces. His work provided a rigorous mathematical foundation for the concept. - **Impact on Quantum Mechanics:** While he didn't directly work on quantum mechanics, Hilbert spaces became instrumental in its mathematical formulation, notably by John von Neumann. - **Broader Mathematical Influence:** Hilbert spaces now underpin many areas of mathematics, including functional analysis, operator theory, and harmonic analysis. **Other Notable Contributions** - **Hilbert's Problems:** In 1900, Hilbert presented 23 unsolved problems that significantly shaped the direction of 20th-century mathematics. - **Foundations of Geometry:** Hilbert's work "Grundlagen der Geometrie" (Foundations of Geometry) revolutionized how geometry was axiomatized. - **Formalism:** Hilbert was an advocate of a philosophical approach called formalism, which emphasized the importance of rigorous axiomatic systems in mathematics. **Legacy** David Hilbert is widely considered one of the most influential mathematicians of the 19th and early 20th centuries. His emphasis on abstraction, rigor, and the broad applicability of mathematics continues to shape research and applications today. # References ```dataview Table title as Title, authors as Authors where contains(subject, "David Hilbert") or contains(authors, "David Hilbert") or contains(subject, "Square-free Word") or contains(subject, "square-free words") sort modified desc, authors, title ```