Leonidas Alaoglu
Encyclopedia
Leonidas Alaoglu was a Canadian-American
mathematician
, most famous for his widely-cited result called Alaoglu's theorem on the weak-star
compactness of the closed unit ball
in the dual
of a normed space, also known as the Banach–Alaoglu theorem.
to Greek
parents. He received his BS
in 1936, Master's in 1937, and PhD in 1938 (at the age of 24), all from the University of Chicago
. His thesis, written under the direction of Lawrence M. Graves was entitled Weak topologies of normed linear spaces. His doctoral thesis is the source of Alaoglu's theorem. The Bourbaki–Alaoglu theorem is a generalization of this result by Bourbaki to dual topologies
.
After some years teaching at Pennsylvania State College
, Harvard University
and Purdue University
, in 1944 he became an operations analyst
for the United States Air Force
. In his last position, from 1953 to 1981 he worked as a senior scientist in operations research at the Lockheed Corporation
in Burbank, California
. In this latter period he wrote numerous research reports, some of them classified.
During the Lockheed years he took an active part in seminar
s and other mathematical activities at Caltech, UCLA and USC
. After his death in 1981 a Leonidas Alaoglu Memorial Lecture Series was established at Caltech. Speakers have included Paul Erdős
, Irving Kaplansky
, Paul Halmos
and Hugh Woodin.
|doi= 10.2307/1968829 | jstor = 1968829 }} |doi= 10.2307/2306079 | issue= 1 | jstor = 2306079 }} | doi = 10.2307/1990319 | format = – Scholar search | issue = 3 | jstor = 00029947/ }} | doi = 10.1090/S0002-9904-1944-08257-8 | issue = 12 }} | doi = 10.2307/1969004 | format = – Scholar search | publisher = The Annals of Mathematics, Vol. 41, No. 2 | jstor = 0003486x }}
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, most famous for his widely-cited result called Alaoglu's theorem on the weak-star
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
compactness of the closed unit ball
Unit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
in the dual
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of a normed space, also known as the Banach–Alaoglu theorem.
Life and work
Alaoglu was born in Red Deer, AlbertaRed Deer, Alberta
Red Deer is a city in Central Alberta, Canada. It is located near the midpoint of the Calgary-Edmonton Corridor and is surrounded by Red Deer County. It is Alberta's third-most-populous city – after Calgary and Edmonton. The city is located in aspen parkland, a region of rolling hills...
to Greek
Greeks
The Greeks, also known as the Hellenes , are a nation and ethnic group native to Greece, Cyprus and neighboring regions. They also form a significant diaspora, with Greek communities established around the world....
parents. He received his BS
Bachelor of Science
A Bachelor of Science is an undergraduate academic degree awarded for completed courses that generally last three to five years .-Australia:In Australia, the BSc is a 3 year degree, offered from 1st year on...
in 1936, Master's in 1937, and PhD in 1938 (at the age of 24), all from the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
. His thesis, written under the direction of Lawrence M. Graves was entitled Weak topologies of normed linear spaces. His doctoral thesis is the source of Alaoglu's theorem. The Bourbaki–Alaoglu theorem is a generalization of this result by Bourbaki to dual topologies
Dual topology
In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space....
.
After some years teaching at Pennsylvania State College
Pennsylvania State University
The Pennsylvania State University, commonly referred to as Penn State or PSU, is a public research university with campuses and facilities throughout the state of Pennsylvania, United States. Founded in 1855, the university has a threefold mission of teaching, research, and public service...
, Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
and Purdue University
Purdue University
Purdue University, located in West Lafayette, Indiana, U.S., is the flagship university of the six-campus Purdue University system. Purdue was founded on May 6, 1869, as a land-grant university when the Indiana General Assembly, taking advantage of the Morrill Act, accepted a donation of land and...
, in 1944 he became an operations analyst
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...
for the United States Air Force
United States Air Force
The United States Air Force is the aerial warfare service branch of the United States Armed Forces and one of the American uniformed services. Initially part of the United States Army, the USAF was formed as a separate branch of the military on September 18, 1947 under the National Security Act of...
. In his last position, from 1953 to 1981 he worked as a senior scientist in operations research at the Lockheed Corporation
Lockheed Corporation
The Lockheed Corporation was an American aerospace company. Lockheed was founded in 1912 and later merged with Martin Marietta to form Lockheed Martin in 1995.-Origins:...
in Burbank, California
Burbank, California
Burbank is a city in Los Angeles County in Southern California, United States, north of downtown Los Angeles. The estimated population in 2010 was 103,340....
. In this latter period he wrote numerous research reports, some of them classified.
During the Lockheed years he took an active part in seminar
Seminar
Seminar is, generally, a form of academic instruction, either at an academic institution or offered by a commercial or professional organization. It has the function of bringing together small groups for recurring meetings, focusing each time on some particular subject, in which everyone present is...
s and other mathematical activities at Caltech, UCLA and USC
University of Southern California
The University of Southern California is a private, not-for-profit, nonsectarian, research university located in Los Angeles, California, United States. USC was founded in 1880, making it California's oldest private research university...
. After his death in 1981 a Leonidas Alaoglu Memorial Lecture Series was established at Caltech. Speakers have included Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
, Irving Kaplansky
Irving Kaplansky
Irving Kaplansky was a Canadian mathematician.-Biography:He was born in Toronto, Ontario, Canada, after his parents emigrated from Poland and attended the University of Toronto as an undergraduate. After receiving his Ph.D...
, Paul Halmos
Paul Halmos
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
and Hugh Woodin.
See also
- Axiom of Choice – The Banach-Alaoglu theorem is not provable from ZF without use of the Axiom of Choice.
- Banach–Alaoglu theorem
- Gelfand representationGelfand representationIn mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
- List of functional analysis topics
- Superabundant number – Article explains the 1944 results of Alaoglu and Erdős on this topic
- Tychonoff's theoremTychonoff's theoremIn mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
- Weak topologyWeak topologyIn mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
– Leads to the weak-star topology to which the Banach-Alaoglu theorem applies.
Publications
- Alaoglu, Leonidas (M.S. thesis, U. of Chicago, 1937). "The asymptotic Waring problem for fifth and sixth powers" (24 pages). Advisor: Leonard Eugene DicksonLeonard Eugene DicksonLeonard Eugene Dickson was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory.-Life:Dickson considered himself a Texan by...
- Alaoglu, Leonidas (Ph.D. thesis, U. of Chicago, 1938). "Weak topologies of normed linear spaces" Advisor: Lawrence Graves
|doi= 10.2307/1968829 | jstor = 1968829 }} |doi= 10.2307/2306079 | issue= 1 | jstor = 2306079 }} | doi = 10.2307/1990319 | format = – Scholar search | issue = 3 | jstor = 00029947/ }} | doi = 10.1090/S0002-9904-1944-08257-8 | issue = 12 }} | doi = 10.2307/1969004 | format = – Scholar search | publisher = The Annals of Mathematics, Vol. 41, No. 2 | jstor = 0003486x }}