He introduced the idea of well-posed problem in the theory of partial differential equations. He also gave his name to the Hadamard inequality on volumes, and the Hadamard matrix, on which the Hadamard transform is based. The Hadamard gate in quantum computing uses this matrix.
In his book Psychology of Invention in the Mathematical Field, Hadamard uses introspection to describe mathematical thought processes. In sharp opposition to authors who identify language and cognition, he describes his own mathematical thinking as largely wordless, often accompanied by mental images that condense the overall idea of a proof.
One of his students was André Weil.
- Hadamard, Jacques, "Psychology of Invention in the Mathematical Field". Dover Pubns; November 1990. ISBN 0486201074
- O'Connor, John J., and Edmund F. Robertson. "Jacques Hadamard". MacTutor History of Mathematics archive.
- Life and Work of Jacques Hadamard, Vladimir Maz'ya & T. O. Shaposhnikova, American Mathematical Society, February 1998, hardcover, 574 pages, ISBN 0821808419