## 2.1 Stefan Banach

In 1892, one of the most outstanding mathematicians of the 20th century, a brilliant self-taught scientist, a man of legend - Stefan Banach - was born in Cracow. He was interested in mathematics already in junior high school but treated it rather as a hobby. A series of coincidences determined him to become a leading representative of the Lviv school of mathematics.

In high school, Banach became friends with his schoolmate Witold Wilkosz (a mathematician, physicist, philosopher, and prominent popularizer of science), later a professor of mathematics at Jagiellonian University, with whom he discussed various mathematical puzzles. Later, Otto Nikodym joined this group of mathematical enthusiasts, and it was one of the conversations about mathematics between Banach and Nikodym that led to his meeting with Hugo Steinhaus. Steinhaus recognized the young Banach’s immense mathematical talent and created the conditions that quickly led to the development of one of the most creative, original mathematical minds.

In 1920, thanks to the efforts of Hugo Steinhaus, Banach obtained a position as an assistant at the Lviv Polytechnic under Antoni Łomnicki (a mathematician mainly associated with the Lviv Polytechnic, where he was a professor, head of the Department of Mathematics, dean, and pro-rector). Mathematical talent and hard work soon paid off. In his doctoral thesis, Banach included new, currently fundamental theorems of the emerging discipline of mathematics - functional analysis. One of the results was the definition of a `B-space,’ which today is called a Banach space. *A Banach space is a linear, normed space in which the metric determined by the norm is complete. The definition of these properties is beyond the scope of this book, but we will write something more about linearity and completeness in this chapter.*

There are many anecdotes about Banach’s life. One of them concerns obtaining a doctoral degree. According to the story, Banach did not care about academic titles so much that he did not even intend to seek a doctorate. Seeing the mathematician’s talent and believing that a doctoral degree was necessary, his superiors hatched an intrigue. They assembled his dissertation from loose notes with theorems and dragged him to the doctoral exam by trickery, saying that a delegation from Warsaw had come with some interesting mathematical problems, and the solutions needed to be explained to them. In reality, Banach graduated with a standard doctorate, but his dismissive attitude toward titles was fodder for anecdotes.

Such a brilliant mind attracted other brilliant minds. Back in 1919, Banach co-founded the Mathematical Society in Cracow, which later became the Polish Mathematical Society, and still exists today.

Back in Lviv, he and Steinhaus started the Lviv school of mathematics, which specialized in functional analysis. It was a talent forge, enough to name a few of its representatives: Stanisław Ulam (a co-inventor of the thermonuclear bomb and the Monte Carlo algorithm), Władysław Orlicz (a mathematician, a researcher on function spaces), Mark Kac (a mathematician, probabilist) or Stanisław Mazur (a mathematician developing theories of topological linear spaces).

Stefan Banach’s favorite workplace was the Scottish Café (Polish: Kawiarnia Szkocka), where he would meet with other mathematicians over coffee, cognac, or music to work on problems that fascinated them. These problems were initially written down on napkins or a table until Łucja, Banach’s wife, equipped the group with a thick notebook. This notebook was the legendary Scottish Book – the notebook in which Lviv mathematicians wrote down mathematical problems to solve. Solving more difficult problems was rewarded with prizes. The prizes were as original as the Lviv school as a whole. Suffice it to say that one of them was a live goose. To this day, not all problems have been solved.

Banach’s name is found in many mathematical theorems or other amazing results. One of them is the Banach-Tarski paradox, a theorem stating that a three-dimensional ball can be cut into a finite number of parts from which two ball of the same size can be assembled after appropriate rotation and translation. An amazingly ingenious construction, seemingly impossible. How to double the volume of a ball with rotations? It turns out that it is enough to make the subsets unmeasurable, and you can already do mathematical miracles with them.

Banach’s life is full of amazing twists and turns. Before World War II, he was a respected professor of mathematics. One day John von Neumann came to Lviv to bring Banach to the United States. He was to hand Banach a check with the number 1 written on it and declare that he could add as many zeros as he saw fit. To this, Banach was to reply that *that’s not enough to leave Poland.* But the war deprived him and many other researchers of the possibility of gainful employment. He spent part of the war as a lice feeder at the Typhus Research Institute under Professor Rudolf Weigl – a Polish biologist, inventor of the world’s first effective vaccine against spotted fever. It is estimated that he saved more than 5,000 people during the war. This was not a dream job, although it was precious because it allowed him to avoid some of the repression used by the occupiers.

Steinhaus once said of Banach that he *combined a spark of genius with an inner compulsion that incessantly reminded him of the words of the poet ,,There is only one thing: the ardent glory of craft’’ – and mathematicians know that their craft and that of the poets share the same mystery…*