Per Enflo

Schauder bases were described by Juliusz Schauder in 1927.

Let V denote a Banach space over the field F.

A Schauder basis is a sequence (bn) of elements of V such that for every element v ∈ V there exists a unique sequence (αn) of elements in F so that

where the convergence is understood with respect to the norm topology.

Schauder bases can also be defined analogously in a general topological vector space.

Banach and other Polish mathematicians would work on mathematical problems at the Scottish Café.

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.

Per H. Enflo (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:

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.

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.

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.

However, such embedding techniques have limitations, as shown by Enflo's (1969) theorem:

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."

A uniformly convex space is a Banach space so that, for every \epsilon>0 there is some \delta>0 so that for any two vectors with \|x\|\le1 and \|y\|\le 1,

implies that

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".

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 Stanislaw 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.

The basis problem was posed by Stefan Banach in his book, Theory of Linear Operators.

Banach asked whether every separable Banach space has a Schauder basis.

A Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; 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 spaces including Banach spaces.

Some of the results of these papers are described in Enflo (1976) and in the last chapter of Benyamini and Lindenstrauss.

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; c.f. 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.