In mathematics, a de Branges space (sometimes written De Branges space) is a concept in functional analysis and is constructed from a de Branges function.
The concept is named after Louis de Branges who proved numerous results regarding these spaces, especially as Hilbert spaces, and used those results to prove the Bieberbach conjecture.
De Branges functions.
A Hermite-Biehler function, also known as de Branges function is an entire function "E" from formula_1 to formula_1 that satisfies the inequality formula_3, for all "z" in the upper half of the complex plane formula_4.
Definition 1.
Given a Hermite-Biehler function "E", the de Branges space "B"("E") is defined as the set of all entire functions "F" such that
formula_5
where:
Definition 2.
A de Branges space can also be defined as all entire functions "F" satisfying all of the following conditions:
Definition 3.
There exists also an axiomatic description, useful in operator theory.
As Hilbert spaces.
Given a de Branges space "B"("E"). Define the scalar product:
formula_11
A de Branges space with such a scalar product can be proven to be a Hilbert space.