1. Hilbert space: Hilbert put forward the concept of Hilbert space, which is an infinite vector space with inner product. Hilbert space is widely used in quantum mechanics, functional analysis and other fields.
2. Hilbert transform: Hilbert put forward Hilbert transform, which is a method to convert real number sequence into complex number sequence, and is widely used in signal processing and communication fields.
3. Hilbert problem: Hilbert put forward 23 unsolved mathematical problems, which are called "Hilbert problems". These problems have had a far-reaching impact on mathematical research in the 20th century, and many mathematicians are trying to solve these problems.
4. Incompleteness theorem: Hilbert proved the incompleteness theorem of mathematical system, that is, there are unprovable propositions in any axiomatic system containing natural numbers. This result has a far-reaching impact on the basic theory of mathematics.
5. Hilbert-Bernards Theorem: Hilbert and Bernards cooperated to prove Hilbert-Bernards Theorem, which is an important result in algebraic number theory.
6. Hilbert curve: Hilbert proposed Hilbert curve, which is a continuous but differentiable curve. Hilbert curve has important applications in geometry and topology.
7. Hilbert method: Hilbert put forward a systematic method to solve mathematical problems, emphasizing formalization and rigor, which had a far-reaching impact on the research methods of modern mathematics.