Existenztheorem

Existenztheorem, in mathematics, is a statement that asserts the existence of an object with certain properties under given hypotheses. It guarantees that something exists, without necessarily providing a method to construct the object. Existence theorems are distinguished from uniqueness theorems and from constructive results that not only prove existence but also show how to obtain the object.

Proofs of existence come in constructive and non-constructive forms. Constructive proofs supply an explicit method or algorithm for producing the object. Non-constructive proofs establish existence indirectly, for example by contradiction, or using principles such as the axiom of choice or completeness. In many areas of analysis and topology, non-constructive existence results are common, though some mathematicians prefer constructive proofs for their algorithmic content.

Typical examples of existence theorems include:

- The Fundamental Theorem of Algebra, which states that every non-constant polynomial with complex coefficients has a complex root.

- Brouwer’s fixed point theorem, which asserts that any continuous function from a compact convex subset of Euclidean space to itself has a fixed point.

- The Picard–Lindelöf theorem, which guarantees the existence (and under further conditions, uniqueness) of solutions to certain initial value problems for ordinary differential equations.

- Bolzano–Weierstrass theorem, which ensures the existence of a convergent subsequence in every bounded sequence.

- Existence of maximal ideals in a ring, typically proven using Zorn’s lemma.

In German-speaking contexts, the term Existenztheorem is closely related to Existenzsatz, both conveying the general idea of asserting existence rather than construction. See also existence proofs, constructive mathematics, and related existence results across branches of mathematics.