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In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. [1] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" [2] or "∃ =1". For example, the formal statement
A standard proof relies on transforming the differential equation into an integral equation, then applying the Banach fixed-point theorem to prove the existence of a solution, and then applying Grönwall's lemma to prove the uniqueness of the solution.
The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by André Weil. [4] Weil's proof used the axiom of choice and Henri Cartan furnished a proof that avoided its use. [5] Cartan's proof also establishes the existence and the uniqueness simultaneously.
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. [2]
In particular, it provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem. It is named for Thomas Hakon Grönwall (1877–1932). Grönwall is the Swedish spelling of his name, but he spelled his name as Gronwall in his scientific publications after ...
A uniqueness theorem (or its proof) is, at least within the mathematics of differential equations, often combined with an existence theorem (or its proof) to a combined existence and uniqueness theorem (e.g., existence and uniqueness of solution to first-order differential equations with boundary condition). [3]
Section 14.4 The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map from L 2 (R n) to the Segal–Bargmann space that intertwines the usual annihilation and creation operators with the operators a j and a ∗ j. This unitary map is the Segal–Bargmann transform.
Proof of the Great Picard Theorem Suppose f is an analytic function on the punctured disk of radius r around the point w , and that f omits two values z 0 and z 1 . By considering ( f ( p + rz ) − z 0 )/( z 1 − z 0 ) we may assume without loss of generality that z 0 = 0, z 1 = 1, w = 0, and r = 1.