In
mathematics, specifically
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conver ...
and
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 (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, the Kirszbraun theorem states that if is a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of some
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, and is another Hilbert space, and
:
is a
Lipschitz-continuous map, then there is a Lipschitz-continuous map
:
that extends and has the same Lipschitz constant as .
Note that this result in particular applies to
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
s and , 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 is a
separable space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of t ...
(in particular, if it is a Euclidean space) the result is true in
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...
; for the fully general case, it appears to need some form of the axiom of choice; the
Boolean prime ideal theorem
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by consid ...
is known to be sufficient.
The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vec ...
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
with the
maximum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
and
carries the Euclidean norm. More generally, the theorem fails for
equipped with any
norm (
) (Schwartz 1969, p. 20).
Explicit formulas
For an
-valued function the extension is provided by
where
is the Lipschitz constant of
on .
In general, an extension can also be written for
-valued functions as
where
and conv(''g'') is the lower convex envelope of ''g''.
History
The theorem was proved by
Mojżesz David Kirszbraun, 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.
References
External links
''Kirszbraun theorem''at
Encyclopedia of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics.
Overview
The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...
.
{{DEFAULTSORT:Kirszbraun Theorem
Lipschitz maps
Metric geometry
Theorems in real analysis
Theorems in functional analysis
Hilbert space