Per Enflo

Per H. Enflo (Swedish: [ˈpæːr ˈěːnfluː]; born 20 May 1944) is a Swedish mathematician working primarily in functional analysis, a field in which he solved problems that had been considered fundamental. Three of these problems had been open for more than forty years:

The basis problem and the approximation problem and later the invariant subspace problem for Banach spaces. In solving these problems, Enflo developed new techniques which were then used by other researchers in functional analysis and operator theory for years. Some of Enflo's research has been important also in other mathematical fields, such as number theory, and in computer science, especially computer algebra and approximation algorithms. Enflo works at Kent State University, where he holds the title of University Professor. Enflo has earlier held positions at the Miller Institute for Basic Research in Science at the University of California, Berkeley, Stanford University, École Polytechnique, (Paris) and The Royal Institute of Technology, Stockholm. Enflo is also a concert pianist.

Enflo's contributions to functional analysis and operator theory In mathematics, functional analysis is concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. In functional analysis, an important class of vector spaces consists of the complete normed vector spaces over the real or complex numbers, which are called Banach spaces. An important example of a Banach space is a Hilbert space, where the norm arises from an inner product. Hilbert spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics, stochastic processes, and time-series analysis. Besides studying spaces of functions, functional analysis also studies the continuous linear operators on spaces of functions.

Hilbert's fifth problem and embeddings

At Stockholm University, Hans Rådström suggested that Enflo consider Hilbert's fifth problem 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. Algorithm theorists derive approximation algorithms that embed finite metric spaces into low-dimensional Euclidean spaces with low "distortion" (in Gromov's terminology for the Lipschitz category; cf. Banach–Mazur distance). Low-dimensional problems have lower computational complexity, of course. More importantly, if the problems embed well in either the Euclidean plane or the three-dimensional Euclidean space, then geometric algorithms become exceptionally fast. However, such embedding techniques have limitations, as shown by Enflo's (1969) theorem:

For every m ≥ 2 {\displaystyle m\geq 2} , the Hamming cube C m {\displaystyle C_{m}} cannot be embedded with "distortion D {\displaystyle D} " (or less) into 2 m {\displaystyle 2^{m}} -dimensional Euclidean space if D < m {\displaystyle D<{\sqrt {m}}} . Consequently, the optimal embedding is the natural embedding, which realizes { 0 , 1 } m {\displaystyle \{0,1\}^{m}} as a subspace of m {\displaystyle m} -dimensional Euclidean space. This theorem, "found by Enflo [1969], is probably the first result showing an unbounded distortion for embeddings into Euclidean spaces. Enflo considered the problem of uniform embeddability among Banach spaces, and the distortion was an auxiliary device in his proof."

Geometry of Banach spaces

A uniformly convex space is a Banach space so that, for every ϵ > 0 {\displaystyle \epsilon >0} there is some δ > 0 {\displaystyle \delta >0} so that for any two vectors with ‖ x ‖ ≤ 1 {\displaystyle \|x\|\leq 1} and ‖ y ‖ ≤ 1 , {\displaystyle \|y\|\leq 1,}

‖ x + y ‖ > 2 − δ {\displaystyle \|x+y\|>2-\delta }

implies that

‖ x − y ‖ < ϵ . {\displaystyle \|x-y\|<\epsilon .}

Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short. In 1972 Enflo proved that "every super-reflexive Banach space 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, the "Goose problem" of Stanisław Mazur, and the approximation problem of Alexander Grothendieck. Grothendieck had shown that his approximation problem was the central problem in the theory of Banach spaces and continuous linear operators.

Basis problem of Banach

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