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STEFAN BANACH
STEFAN BANACH
Founded the important modern
mathematical field of functional analysis
and made major contributions to the
theory of topological vector spaces.
In addition, he contributed to measure
theory, integration, the theory of sets and
orthogonal series.
THE LIFE OF STEFAN BANACH
• Early years – born March 30, 1892
• Gymnasium - tutoring
• “mathematics is too sharp a tool to put into
the hands of children; for training in logical
thinking, there is nothing better than
accusativus cum infinitivo and ablativus
absolutus.”
THE LIFE OF STEFAN BANACH
• Lvov Polytechnic
• “since mathematics was so highly
developed, it would be impossible to do
anything new in this discipline…”
• 1914 – half-diploma examinations
(freshman and sophomore years)
THE LIFE OF STEFAN BANACH
• Word War I- excused from military
services
• Alfred Whitehead and Bernard Russell’s
Principia Mathematica and Einstein’s
special and general theories of relativity
• Lectures at Jagiellonian University
• Hugo Steinhaus’ discovery
• Marriage - Lucja Braus in 1920
THE LIFE OF STEFAN BANACH
• 1920 – doctoral dissertation: “On
operations on abstract sets and their
applications to integral equations”
• Introduces an abstract object that later
came to be called a Banach space
• To some degree, this dissertation brought
functional analysis to independent life.
• Obstacles in formal process of obtaining
Ph.D
THE LIFE OF STEFAN BANACH
• 1922 – Banach received his habilitation,
became a Professor Extraordinarius at Jan
Kazimierz University
• Need for writing textbooks
The Scottish Café
The Scottish Café
• What the Cafés of Montmartre did for the
arts of Fin-de-Siècle Paris, the Scottish
Café did for mathematics in Lvov.
• Incredibly fruitful collaboration of a group
of unusually gifted and original minds.
• …tiny tables with marble tops were
extremely useful as tablets to be covered
with mathematical formulas. At first the
owner was not overly enthusiastic…
The Scottish Café
• We have to regretfully state that many
valuable results of Banach and of his
school were lost (…) as a result of lack of
pedantry among members of the school
and, first of all, Banach himself.
• Scottish Café – a phenomenon of
teamwork in unorthodox places that led to
joint solution of research problems
The Scottish Book
• One of the most revered relics of the
mathematical world
• A regular, ruled school notebook
• An unofficial communal scientific
publication – anyone interested could write
down problems to be solved and anyone
could write his solutions.
The Scottish Book
• Many problems from Scottish Book played
a significant role the development of
functional analysis and other branches of
mathematics.
• Prizes - life goose, bottle of wine, flask of
brandy
• 1972 – life goose was presented to
Swedish mathematician Per Enflo.
THE LIFE OF STEFAN BANACH
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Banach’s unconventional behavior
Superb teacher
Author of elementary textbooks
Soviet occupation – Dean of the PhysicalMathematical Faculty and Head of the
Department of Mathematical Analysis
THE LIFE OF STEFAN BANACH
• German invasion – 1941
• Himmler’s Extraordinary Pacification
Action – 40 Polish scholars professors,
writers, and other distinguished
representatives of Lvov intelligentsia
perished at the hands of the Nazis.
• Banach’s pitiful physical condition – feeder
of lice in the Rudolf Weigl Bacteriological
Institute until July 1944
THE LIFE OF STEFAN BANACH
• An offer of a chair at the Jagiellonian
University
• Minister of Education
• Banach died on August 31 1945 in Lvov.
Banach’s Fixed Point Theorem
A theorem stating that if a
mapping ƒ of a metric space E
into itself is a contraction, then
there exists a unique element x
of E such that ƒ(x) = x. Also
known as contraction mapping
principle.
Banach-Tarski Paradox
• Did you know that it is possible to cut a
solid ball into 5 pieces, and by reassembling them, using rigid motions only,
form TWO solid balls, EACH THE SAME
SIZE AND SHAPE as the original?
• So why can't you do this in real life, say,
with a block of gold?
Banach-Tarski Paradox
• If matter were infinitely divisible (which it is not)
then it might be possible. But the pieces involved
are so "jagged" and exotic that they do not have
a well-defined notion of volume, or measure,
associated to them. In fact, what the BanachTarski paradox shows is that no matter how you
try to define "volume" so that it corresponds with
our usual definition for nice sets, there will
always be "bad" sets for which it is impossible to
define a "volume"! (Or else the above example
would show that 2 = 1.)
Banach-Tarski Paradox
It is interesting to note that one corollary to
This paradox is that you can take a sphere, cut it
into n pieces, remove some of the pieces, and
reassemble the remaining pieces back into the
original sphere without missing anything.
Obviously it is not possible with a physical
sphere; but it is possible with mathematical
spheres (which are infinitely divisible), if the
Axiom of Choice is assumed.
Banach-Tarski Paradox
• An alternate version of this theorem says: it is
possible to take a solid ball the size of a pea,
and by cutting it into a FINITE number of pieces,
reassemble it to form A SOLID BALL THE SIZE
OF THE SUN.
• You might want to say that mathematics in this
case reveals to us that we must be very careful
about how we define things (like volumes) that
seem very intuitive to us.
Banach-Tarski Paradox
• First of all, if we didn't restrict ourselves to
rigid motions, it would be more believable.
For instance, you can take the interval
[0,1], stretch it to twice its length and cut it
into 2 pieces each the same as the original
interval. Secondly, if we didn't restrict
ourselves to a finite number of pieces, it
would be more believable, too: the
cardinality of the number of points in one
ball is the same as that of two balls!
Banach-Tarski Paradox
• Let A be a unit circle, and let B be a unit circle
with one point X missing (called a "deleted
circle"). Are sets A and B equidecomposable?
Consider set B and let U be the subset
consisting of all points that are a positive integer
number of radians clockwise from X along the
circle. This is a countable infinite set (the
irrationality of Pi prevents two such points from
coinciding). Let set V be everything else.
• If you pick set U up and rotate it
counterclockwise by one radian, something very
interesting happens. The deleted hole at X gets
filled by the point 1 radian away, and the point at
the (n-1)-th radian gets filled by the point at the
n-th radian. Every point vacated gets filled, and
in addition, the empty point at X gets filled too!
Banach-Tarski Paradox
• Thus, B may be decomposed into
sets U and V, which after this
reassembling, form set A, a complete
circle!
• This elementary example forms the
beginnings of the idea of how to
accomplish the Banach-Tarski
paradox
High School Connections
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Banach’s biography
Math Fun Facts
Norm and vector’s absolute value
Banach’s Contraction Principle
“Scottish Book”- form of working with
mathematically gifted students
(classroom’s blogs or websites)
REFERENCES
• Roman Kaluza, Through a reporter’s eyes. The Life of
Stefan Banach
• Su, Francis E., et al. "Banach-Tarski Paradox." Mudd
Math Fun Facts. <http://www.math.hmc.edu/funfacts>.
• Su, Francis E., et al. "Equidecomposability." Mudd Math
Fun Facts. <http://www.math.hmc.edu/funfacts>.