Per H. Enflo (; born 20 May 1944) is a Swedish mathematician working primarily in
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 space#Definition, inner product, Norm (mathematics)#Defini ...
, a field in which he solved
problems
A problem is a difficulty which may be resolved by problem solving.
Problem(s) or The Problem may also refer to:
People
* Problem (rapper), (born 1985) American rapper Books
* Problems (Aristotle), ''Problems'' (Aristotle), an Aristotelian (or ps ...
that had been considered fundamental. Three of these problems had been
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
for more than forty years:
* The
basis problem and the
approximation problem
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
and later
* the
invariant subspace problem
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many vari ...
for
Banach spaces
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 vector ...
.
In solving these problems, Enflo developed new techniques which were then used by other researchers in
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 space#Definition, inner product, Norm (mathematics)#Defini ...
and
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operat ...
for years. Some of Enflo's research has been important also in other mathematical fields, such as
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, and in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, especially
computer algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
and
approximation algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solu ...
s.
Enflo works at
Kent State University
Kent State University (KSU) is a public research university in Kent, Ohio. The university also includes seven regional campuses in Northeast Ohio and additional facilities in the region and internationally. Regional campuses are located in As ...
, where he holds the title of University Professor. Enflo has earlier held positions at the
Miller Institute
The Miller Institute for Basic Research in Science was established on the University of California, Berkeley, campus in 1955 after Adolph C. Miller and his wife, Mary Sprague Miller, made a donation to the university. It was their wish that the d ...
for Basic Research in Science at the
University of California, Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
,
Stanford University
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is consider ...
,
École Polytechnique
École may refer to:
* an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing in région Île-de-France
* École, Savoi ...
, (
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. S ...
) and The
Royal Institute of Technology
The KTH Royal Institute of Technology ( sv, Kungliga Tekniska högskolan, lit=Royal Institute of Technology), abbreviated KTH, is a public research university in Stockholm, Sweden. KTH conducts research and education in engineering and technolo ...
,
Stockholm
Stockholm () is the Capital city, capital and List of urban areas in Sweden by population, largest city of Sweden as well as the List of urban areas in the Nordic countries, largest urban area in Scandinavia. Approximately 980,000 people liv ...
.
Enflo is also a
concert pianist
A pianist ( , ) is an individual musician who plays the piano. Since most forms of Western music can make use of the piano, pianists have a wide repertoire and a wide variety of styles to choose from, among them traditional classical music, ja ...
.
Enflo's contributions to functional analysis and operator theory
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
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 space#Definition, inner product, Norm (mathematics)#Defini ...
is concerned with the study of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s and
operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
acting upon them. It has its historical roots in the study of
functional space
In mathematics, a function space is a Set (mathematics), set of function (mathematics), functions between two fixed sets. Often, the Domain of a function, domain and/or codomain will have additional Mathematical structure, structure which is inher ...
s, in particular transformations of
functions, such as the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, as well as in the study of
differential and
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
equations. In functional analysis, an important class of vector spaces consists of the
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s over the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
numbers, which are called
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 vector ...
s. An important example of a Banach space is a
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 ...
, where the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
arises from an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
. Hilbert spaces are of fundamental importance in many areas, including the mathematical formulation of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
stochastic processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
, and
time-series analysis
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
. Besides studying spaces of functions, functional analysis also studies the
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
linear operators
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
on spaces of functions.
Hilbert's fifth problem and embeddings
At Stockholm University, Hans Rådström suggested that Enflo consider
Hilbert's fifth problem
Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups.
The theory of Lie groups describes continuous symmetry in mathemat ...
in the spirit of functional analysis. In two years, 1969–1970, Enflo published five papers on Hilbert's fifth problem; these papers are collected in Enflo (1970), along with a short summary. Some of the results of these papers are described in Enflo (1976) and in the last chapter of Benyamini and
Lindenstrauss.
Applications in computer science
Enflo's techniques have found application in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
.
Algorithm theorists derive
approximation algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solu ...
s that embed finite metric spaces into low-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s with low "distortion" (in
Gromov
Gromov (russian: Громов) is a Russian male surname, its feminine counterpart is Gromova (Громова).
Gromov may refer to:
* Alexander Georgiyevich Gromov (born 1947), Russian politician and KGB officer
* Alexander Gromov (born 1959), R ...
's terminology for the
Lipschitz category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
; c.f.
Banach–Mazur distance). Low-dimensional problems have lower
computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...
, of course. More importantly, if the problems embed well in either the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
or the three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, then
geometric algorithms
The following is a list of well-known algorithms along with one-line descriptions for each.
Automated planning
Combinatorial algorithms
General combinatorial algorithms
* Brent's algorithm: finds a cycle in function value iterations using on ...
become exceptionally fast.
However, such
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is gi ...
techniques have limitations, as shown by Enflo's (1969) theorem:
: For every
, the
Hamming cube
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to ch ...
cannot be embedded with "distortion
" (or less) into
-dimensional Euclidean space if
. Consequently, the optimal embedding is the natural embedding, which realizes
as a subspace of
-dimensional Euclidean space.
This theorem, "found by Enflo
969
Year 969 ( CMLXIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar, the 969th year of the Common Era (CE) and ''Anno Domini'' (AD) designations, the 969th year of the 1st millennium, the 69th ...
is probably the first result showing an unbounded distortion for
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is gi ...
s into
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s. Enflo considered the problem of
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
embeddability among
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 vector ...
s, and the distortion was an auxiliary device in his proof."
Geometry of Banach spaces
A uniformly convex space is a
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 vector ...
so that, for every
there is some
so that for any two vectors with
and
:
implies that
:
Intuitively, the center of a line segment inside the
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
must lie deep inside the unit ball unless the segment is short.
In 1972 Enflo proved that "every
super-reflexive 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 vector ...
admits an equivalent
uniformly convex norm".
The basis problem and Mazur's goose
With one paper, which was published in 1973, Per Enflo solved three problems that had stumped functional analysts for decades: The
basis problem of
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
, the "
Goose problem" of
Stanislaw Mazur, and the
approximation problem
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
of
Alexander Grothendieck. Grothendieck had shown that his approximation problem was the central problem in the
theory
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
of
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 vector ...
s and
continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear o ...
s.
Basis problem of Banach
The basis problem was posed by Stefan Banach in his book, ''Theory of Linear Operators''. Banach asked whether every separable
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 vector ...
has a
Schauder basis In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...
.
A Schauder basis or countable basis is similar to the usual (Hamel)
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
; the difference is that for Hamel bases we use
linear combinations that are ''finite'' sums, while for Schauder bases they may be ''infinite'' sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s including
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 vector ...
s.
Schauder bases were described by
Juliusz Schauder
Juliusz Paweł Schauder (; 21 September 1899, Lwów, Austria-Hungary – September 1943, Lwów, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equations and mathema ...
in 1927. Let ''V'' denote a
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 vector ...
over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''F''. A ''Schauder basis'' is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
(''b''
''n'') of elements of ''V'' such that for every element ''v'' ∈ ''V'' there exists a ''unique'' sequence (α
''n'') of elements in ''F'' so that
:
where the
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four Wei ...
is understood with respect to the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. Schauder bases can also be defined analogously in a general
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
.
Problem 153 in the Scottish Book: Mazur's goose
![MazurGes](https://upload.wikimedia.org/wikipedia/commons/e/ec/MazurGes.jpg)
Banach and other Polish mathematicians would work on mathematical problems at the
Scottish Café
The Scottish Café ( pl, Kawiarnia Szkocka) was a café in Lwów, Poland (now Lviv, Ukraine) where, in the 1930s and 1940s, mathematicians from the Lwów School of Mathematics collaboratively discussed research problems, particularly in functio ...
. When a problem was especially interesting and when its solution seemed difficult, the problem would be written down in the book of problems, which soon became known as the ''
Scottish Book''. For problems that seemed especially important or difficult or both, the problem's proposer would often pledge to award a prize for its solution.
On 6 November 1936,
Stanislaw Mazur posed a problem on representing continuous functions. Formally writing down ''problem 153'' in the ''Scottish Book'', Mazur promised as the reward a "live goose", an especially rich price during the
Great Depression
The Great Depression (19291939) was an economic shock that impacted most countries across the world. It was a period of economic depression that became evident after a major fall in stock prices in the United States. The economic contagio ...
and on the eve of
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...
.
Fairly soon afterwards, it was realized that Mazur's problem was closely related to Banach's problem on the existence of Schauder bases in separable Banach spaces. Most of the other problems in the ''Scottish Book'' were solved regularly. However, there was little progress on Mazur's problem and a few other problems, which became famous
open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is know ...
s to mathematicians around the world.
Grothendieck's formulation of the approximation problem
Grothendieck's work on the
theory
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
of Banach spaces and
continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear o ...
s introduced the
approximation property
In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true.
Every Hilbert space has this property ...
. A
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 vector ...
is said to have the
approximation property
In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true.
Every Hilbert space has this property ...
, if every
compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...
is a limit of
finite-rank operator
In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional.
Finite-rank operators on a Hilbert space A canonical form
Finite-rank operators are ...
s. The converse is always true.
Joram Lindenstrauss
Joram Lindenstrauss ( he, יורם לינדנשטראוס) (October 28, 1936 – April 29, 2012) was an Israeli mathematician working in functional analysis. He was a professor of mathematics at the Einstein Institute of Mathematics.
Biograph ...
and L. Tzafriri.
In a long monograph, Grothendieck proved that if every Banach space had the approximation property, then every Banach space would have a Schauder basis. Grothendieck thus focused the attention of functional analysts on deciding whether every Banach space have the approximation property.
Enflo's solution
In 1972, Per Enflo constructed a separable Banach space that lacks the approximation property and a Schauder basis. In 1972, Mazur awarded
live gooseto Enflo in a ceremony at the
Stefan Banach Center in
Warsaw
Warsaw ( pl, Warszawa, ), officially the Capital City of Warsaw,, abbreviation: ''m.st. Warszawa'' is the capital and largest city of Poland. The metropolis stands on the River Vistula in east-central Poland, and its population is officia ...
; the "goose reward" ceremony was broadcast throughout
Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populous ...
.
Invariant subspace problem and polynomials
In
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 space#Definition, inner product, Norm (mathematics)#Defini ...
, one of the most prominent problems was the
invariant subspace problem
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many vari ...
, which required the evaluation of the truth of the following proposition:
:Given a complex
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 vector ...
''H'' of
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
> 1 and a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
''T'' : ''H'' → ''H'', then ''H'' has a non-trivial
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
''T''-invariant subspace, i.e. there exists a closed
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
''W'' of ''H'' which is different from and ''H'' such that ''T''(''W'') ⊆ ''W''.
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 vector ...
s, the first example of an operator without an invariant subspace was constructed by Enflo. (For
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 ...
s, the
invariant subspace problem
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many vari ...
remains
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
.)
Enflo proposed a solution to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987
[Beauzamy 1988; Yadav.] Enflo's long "manuscript had a world-wide circulation among mathematicians" and some of its ideas were described in publications besides Enflo (1976). Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Beauzamy, who acknowledged Enflo's ideas.
In the 1990s, Enflo developed a "constructive" approach to the
invariant subspace problem
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many vari ...
on Hilbert spaces.
Multiplicative inequalities for homogeneous polynomials
An essential idea in Enflo's construction was "concentration of polynomials at low degrees": For all positive integers
and
, there exists
such that for all
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
s
and
of degrees
and
(in
variables), then
where
denotes the sum of the absolute values of the coefficients of
. Enflo proved that
does not depend on the number of variables
. Enflo's original proof was simplified by
Montgomery.
This result was generalized to other
norms on the vector space of
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
s. Of these norms, the most used has been the
Bombieri norm In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in \mathbb R or \mathbb C (there is also a version for non homogeneous univariate polynomials). This norm has many remarkable prop ...
.
= Bombieri norm
=
The
Bombieri norm In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in \mathbb R or \mathbb C (there is also a version for non homogeneous univariate polynomials). This norm has many remarkable prop ...
is defined in terms of the following
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
:
For all
we have
:
if
: For every
we define
where we use the following notation:
if
, we write
and
and
The most remarkable property of this norm is the Bombieri inequality:
Let
be two
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
s respectively of degree
and
with
variables, then, the following inequality holds:
:
In the above statement, the Bombieri inequality is the left-hand side inequality; the right-hand side inequality means that the
Bombieri norm In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in \mathbb R or \mathbb C (there is also a version for non homogeneous univariate polynomials). This norm has many remarkable prop ...
is a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
of the
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
of polynomials under multiplication.
The Bombieri inequality implies that the product of two polynomials cannot be arbitrarily small, and this lower-bound is fundamental in applications like
polynomial factorization
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field (mathematics), field or in the integers as the product of irreducible polynomial, irreducible ...
(or in Enflo's construction of an operator without an invariant subspace).
Applications
Enflo's idea of "concentration of polynomials at low degrees" has led to important publications in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
algebraic and Diophantine geometry, and
polynomial factorization
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field (mathematics), field or in the integers as the product of irreducible polynomial, irreducible ...
.
Mathematical biology: Population dynamics
In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, Per Enflo has published several papers in
mathematical biology
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development a ...
, specifically in
population dynamics
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.
History
Population dynamics has traditionally been the dominant branch of mathematical biology, which has ...
.
Human evolution
Enflo has also published in
population genetics
Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and pop ...
and
paleoanthropology
Paleoanthropology or paleo-anthropology is a branch of paleontology and anthropology which seeks to understand the early development of anatomically modern humans, a process known as hominization, through the reconstruction of evolutionary kinship ...
.
Today, all humans belong to one population of ''Homo sapiens sapiens'', which is individed by species barrier. However, according to the "Out of Africa" model this is not the first species of hominids: the first species of genus ''Homo'', ''Homo habilis'', evolved in East Africa at least 2 Ma, and members of this species populated different parts of Africa in a relatively short time. ''Homo erectus'' evolved more than 1.8 Ma, and by 1.5 Ma had spread throughout the
Old World
The "Old World" is a term for Afro-Eurasia that originated in Europe , after Europeans became aware of the existence of the Americas. It is used to contrast the continents of Africa, Europe, and Asia, which were previously thought of by the ...
.
Anthropologists have been divided as to whether current human population evolved as one interconnected population (as postulated by the
Multiregional Evolution The multiregional hypothesis, multiregional evolution (MRE), or polycentric hypothesis is a scientific model that provides an alternative explanation to the more widely accepted "Out of Africa" model of monogenesis for the pattern of human evoluti ...
hypothesis), or evolved only in East Africa,
speciated, and then migrating out of Africa and replaced human populations in
Eurasia
Eurasia (, ) is the largest continental area on Earth, comprising all of Europe and Asia. Primarily in the Northern and Eastern Hemispheres, it spans from the British Isles and the Iberian Peninsula in the west to the Japanese archipelago a ...
(called the "Out of Africa" Model or the "Complete Replacement" Model).
Neanderthals and modern humans coexisted in Europe for several thousand years, but the duration of this period is uncertain. Modern humans may have first migrated to Europe 40–43,000 years ago.
Neanderthals may have lived as recently as 24,000 years ago in
refugia on the south coast of the Iberian peninsula such as
Gorham's Cave
Gorham's Cave ( es, Cueva de Gorham, ) is a sea-level cave in the British overseas territory of Gibraltar. Though not a sea cave, it is often mistaken for one. Considered to be one of the last known habitations of the Neanderthals in Europe, the ...
. Inter-stratification of Neanderthal and modern human remains has been suggested, but is disputed.
With
Hawks
Hawks are birds of prey of the family Accipitridae. They are widely distributed and are found on all continents except Antarctica.
* The subfamily Accipitrinae includes goshawks, sparrowhawks, sharp-shinned hawks and others. This subfamily a ...
and
Wolpoff, Enflo published an explanation of fossil evidence on the
DNA of
Neanderthal
Neanderthals (, also ''Homo neanderthalensis'' and erroneously ''Homo sapiens neanderthalensis''), also written as Neandertals, are an extinct species or subspecies of archaic humans who lived in Eurasia until about 40,000 years ago. While th ...
and
modern human
Early modern human (EMH) or anatomically modern human (AMH) are terms used to distinguish ''Homo sapiens'' (the only extant Hominina species) that are anatomically consistent with the range of phenotypes seen in contemporary humans from extin ...
s. This article tries to resolve a debate in the
evolution of modern humans between theories suggesting either
multiregional and
single African origins. In particular,
the
extinction of Neanderthals could have happened due to waves of modern humans entered Europe – in technical terms, due to "the continuous influx of modern human DNA into the Neandertal gene pool."
Enflo has also written about the population dynamics of
zebra mussels
The zebra mussel (''Dreissena polymorpha'') is a small freshwater mussel. The species originates from the lakes of southern Russia and Ukraine, but has been accidentally Introduced species, introduced to numerous other areas and has become an inv ...
in
Lake Erie
Lake Erie ( "eerie") is the fourth largest lake by surface area of the five Great Lakes in North America and the eleventh-largest globally. It is the southernmost, shallowest, and smallest by volume of the Great Lakes and therefore also has t ...
.
Piano
Per Enflo is also a
concert pianist
A pianist ( , ) is an individual musician who plays the piano. Since most forms of Western music can make use of the piano, pianists have a wide repertoire and a wide variety of styles to choose from, among them traditional classical music, ja ...
.
A
child prodigy
A child prodigy is defined in psychology research literature as a person under the age of ten who produces meaningful output in some domain at the level of an adult expert. The term is also applied more broadly to young people who are extraor ...
in both music and mathematics, Enflo won the Swedish competition for young pianists at age 11 in 1956, and he won the same competition in 1961. At age 12, Enflo appeared as a soloist with the Royal Opera Orchestra of Sweden. He debuted in the
Stockholm Concert Hall
The Stockholm Concert Hall ( sv, Stockholms konserthus) is the main hall for orchestral music in Stockholm, Sweden.
With a design by Ivar Tengbom chosen in competition, inaugurated in 1926, the Hall is home to the Royal Stockholm Philharmoni ...
in 1963. Enflo's teachers included
Bruno Seidlhofer
Bruno Georg Seidlhofer (5 September 1905 – 19 February 1982) was an Austrian pianist, organist, academic teacher and piano teacher at the University of Music and Performing Arts Vienna.
Life
Born in Vienna, Seidlhofer taught piano at the Aca ...
,
Géza Anda
Géza Anda (; 19 November 192113 June 1976) was a Swiss- Hungarian pianist. A celebrated interpreter of classical and romantic repertoire, particularly noted for his performances and recordings of Mozart, he was also considered to be a tremendou ...
, and Gottfried Boon (who himself was a student of Arthur Schnabel).
Chagrin Valley Chamber Music Concert Series 2009-2010
.
In 1999 Enflo competed in the first annual
Van Cliburn Foundation
{{refimprove, date=October 2018
The Van Cliburn Foundation presents the Van Cliburn International Piano Competition, the Cliburn Amateur Piano Competition, the Cliburn International Junior Piano Competition and Festival, ''Musical Awakenings'' educ ...
's
International Piano Competition fo
Outstanding Amateurs[
]
Enflo performs regularly around
Kent
Kent is a county in South East England and one of the home counties. It borders Greater London to the north-west, Surrey to the west and East Sussex to the south-west, and Essex to the north across the estuary of the River Thames; it faces ...
and in a
Mozart
Wolfgang Amadeus Mozart (27 January 17565 December 1791), baptised as Joannes Chrysostomus Wolfgangus Theophilus Mozart, was a prolific and influential composer of the Classical period (music), Classical period. Despite his short life, his ra ...
series in
Columbus, Ohio
Columbus () is the state capital and the most populous city in the U.S. state of Ohio. With a 2020 census population of 905,748, it is the 14th-most populous city in the U.S., the second-most populous city in the Midwest, after Chicago, and t ...
(with the Triune Festival Orchestra). His solo piano recitals have appeared on the Classics Network of the radio station
WOSU, which is sponsored by
Ohio State University
The Ohio State University, commonly called Ohio State or OSU, is a public land-grant research university in Columbus, Ohio. A member of the University System of Ohio, it has been ranked by major institutional rankings among the best publ ...
.
References
Notes
*
Recipients of 2005 Distinguished Scholar Award at Kent State University Announced, ''eInside'', 2005-4-11. Retrieved on February 4, 2007.
Bibliography
* Enflo, Per. (1970) Investigations on Hilbert's fifth problem for non locally compact groups (Stockholm University). Enflo's thesis contains reprints of exactly five papers:
**
**
**
**
**
*** Enflo, Per. 1976
Uniform homeomorphisms between Banach spaces ''Séminaire Maurey-Schwartz (1975—1976), Espaces,
, applications radonifiantes et géométrie des espaces de Banach'', Exp. No. 18, 7 pp. Centre Math., École Polytech., Palaiseau. (57 #17222) [Highlights of papers on
Hilbert's fifth problem
Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups.
The theory of Lie groups describes continuous symmetry in mathemat ...
and on independent results of Martin Ribe, another student of Hans Rådström]
*
*
*
*
*
* (accessible to readers with undergraduate mathematics)
* P. Enflo, John D. Hawks, Milford H. Wolpoff, M. Wolpoff. "A simple reason why Neanderthal ancestry can be consistent with current DNA information". ''American Journal Physical Anthropology'', 2001
*
*
*
*
*
*
*
Grothendieck, A.: ''Produits tensoriels topologiques et espaces nucleaires''. Memo. Amer. Math. Soc. 16 (1955).
*
*
Paul R. Halmos, "Has progress in mathematics slowed down?" ''Amer. Math. Monthly'' 97 (1990), no. 7, 561–588.
* William B. Johnson "Complementably universal separable Banach spaces" in
Robert G. Bartle
Robert Gardner Bartle (November 20, 1927 – September 18, 2003) was an American mathematician specializing in real analysis. He is known for writing the popular textbooks ''The Elements of Real Analysis'' (1964), ''The Elements of Integration'' ...
(ed.), 1980 ''Studies in functional analysis'', Mathematical Association of America.
*
*
* Kwapień, S. "On Enflo's example of a Banach space without the approximation property". Séminaire Goulaouic-Schwartz 1972—1973: Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 8, 9 pp. Centre de Math., École Polytech., Paris, 1973.
*
Lindenstrauss, Joram and Benyamini, Yoav. ''Geometric nonlinear functional analysis'' Colloquium publications, 48. American Mathematical Society.
*
Lindenstrauss, J.
Joram Lindenstrauss ( he, יורם לינדנשטראוס) (October 28, 1936 – April 29, 2012) was an Israeli mathematician working in functional analysis. He was a professor of mathematics at the Einstein Institute of Mathematics.
Biograph ...
; Tzafriri, L.: ''Classical Banach Spaces I, Sequence spaces'', 1977. Springer-Verlag.
* .
*
*
*
*
*
*
Karen Saxe
Karen Saxe is an American mathematician who specializes in functional analysis, and in the mathematical study of issues related to social justice. She is DeWitt Wallace Professor of Mathematics, Emerita at Macalester College,. She is Associate Exec ...
, ''Beginning Functional Analysis'',
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics (UTM) (ISSN 0172-6056) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow boo ...
, 2002 Springer-Verlag, New York. (Pages 122–123 sketch a biography of Per Enflo.)
*
Schmidt, Wolfgang M. (1980
996 with minor corrections ''
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by r ...
.'' Lecture Notes in Mathematics 785. Springer.
* Singer, Ivan. ''Bases in Banach spaces. II''. Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. viii+880 pp. .
*
External sources
Biography of Per Enfloat
Canisius College
Canisius College is a private Jesuit college in Buffalo, New York. It was founded in 1870 by Jesuits from Germany and is named after St. Peter Canisius. Canisius offers more than 100 undergraduate majors and minors, and around 34 master's ...
Homepage of Per Enfloa
Kent State University*
Databases
*
*
*
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