Kirszbraun theorem
Encyclopedia
In mathematics
, specifically real analysis
and functional analysis
, the Kirszbraun theorem states that if U is a subset
of some Hilbert space
H1, and H2 is another Hilbert space, and
is a Lipschitz-continuous
map, then there is a Lipschitz-continuous map
that extends f and has the same Lipschitz constant as f.
Note that this result in particular applies to Euclidean space
s En and Em, and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). If H1 is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory
; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem
is known to be sufficient.
The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach space
s is not true in general, not even for finite-dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of Rn with the maximum norm and Rm carries the Euclidean norm. More generally, the theorem fails for
equipped with any 
norm (
) (Schwartz 1969, p. 20).
For an R-valued function the extension is provided by
where 
is f's Lipschitz constant on U.
, and later it was reproved by Frederick Valentine, who first proved it for the Euclidean plane. Sometimes this theorem is also called Kirszbraun–Valentine theorem.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
and functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the Kirszbraun theorem states that if U is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of some Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H1, and H2 is another Hilbert space, and
- f : U → H2
is a Lipschitz-continuous
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
map, then there is a Lipschitz-continuous map
- F: H1 → H2
that extends f and has the same Lipschitz constant as f.
Note that this result in particular applies to Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s En and Em, and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). If H1 is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem
Boolean prime ideal theorem
In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on...
is known to be sufficient.
The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s is not true in general, not even for finite-dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of Rn with the maximum norm and Rm carries the Euclidean norm. More generally, the theorem fails for
equipped with any norm () (Schwartz 1969, p. 20).For an R-valued function the extension is provided by
where is f's Lipschitz constant on U.History
The theorem was proved by Mojżesz David KirszbraunMojżesz David Kirszbraun
Mojżesz David Kirszbraun — Polish mathematician, mostly known for his theorem on extensions of Lipschitz maps.This theorem appears in his master's thesis, defended in Warsaw in 1930.He finished school in 1922....
, and later it was reproved by Frederick Valentine, who first proved it for the Euclidean plane. Sometimes this theorem is also called Kirszbraun–Valentine theorem.