A theorem stating that if a mapping

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STEFAN BANACH
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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.

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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.

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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)

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THE LIFE OF STEFAN BANACH

• Word War I- excused from military services
• Alfred Whitehead and Bernard Russells
• Principia Mathematica and Einsteins special
  and general theories of relativity
• Lectures at Jagiellonian University
• Hugo Steinhaus discovery
• Marriage - Lucja Braus in 1920

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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

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THE LIFE OF STEFAN BANACH

• 1922 Banach received his habilitation, became a
  Professor Extraordinarius at Jan Kazimierz
  University
• Need for writing textbooks

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The Scottish Café
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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

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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

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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.

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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.

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THE LIFE OF STEFAN BANACH

• Banachs unconventional behavior
• Superb teacher
• Author of elementary textbooks
• Soviet occupation Dean of the
  Physical-Mathematical Faculty and Head of the
  Department of Mathematical Analysis

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THE LIFE OF STEFAN BANACH

• German invasion 1941
• Himmlers Extraordinary Pacification Action 40
  Polish scholars professors, writers, and other
  distinguished representatives of Lvov
  intelligentsia perished at the hands of the
  Nazis.
• Banachs pitiful physical condition feeder of
  lice in the Rudolf Weigl Bacteriological
  Institute until July 1944

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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.

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Banachs Fixed Point Theorem
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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.
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Banach-Tarski Paradox

• Did you know that it is possible to cut a solid
  ball into 5 pieces, and by re-assembling 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?

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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
  Banach-Tarski 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.)

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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.

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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.

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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!

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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!

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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

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High School Connections

• Banachs biography
• Math Fun Facts
• Norm and vectors absolute value
• Banachs Contraction Principle
• Scottish Book- form of working with
  mathematically gifted students (classrooms blogs
  or websites)

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REFERENCES

• Roman Kaluza, Through a reporters eyes. The Life
  of Stefan Banach
• Su, Francis E., et al. "Banach-Tarski Paradox."
  Mudd Math Fun Facts. lthttp//www.math.hmc.edu/funf
  actsgt.
• Su, Francis E., et al. "Equidecomposability."
  Mudd Math Fun Facts. lthttp//www.math.hmc.edu/funf
  actsgt.