Existence Theorem

A theorem stating the existence of an object, such as the solution to a problem or equation. Strictly speaking, it need not tell how many such objects there are, nor give hints on how to find them. Some existence theorems give explicit formulas for solutions (e.g., Cramer's rule), others describe in their proofs iteration processes for approaching them (e.g., Bolzano-Weierstrass theorem), while others are settled by nonconstructive proofs which simply deduce the necessity of solutions without indicating any method for determining them (e.g., the Brouwer fixed point theorem, which is proved by reductio ad absurdum, showing that the nonexistence would lead to a contradiction).

See also

Existence Problem, Existential Sentence, Picard's Existence Theorem, Uniqueness Theorem

This entry contributed by Margherita Barile

Explore with Wolfram|Alpha

Cite this as:

Barile, Margherita. "Existence Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications