Stefan Banach

How to use this course

Stefan Banach turned a scattered set of nineteenth-century results about integral equations, Fourier series and function spaces into one theory. Its objects are vector spaces with a notion of length in which Cauchy sequences converge. Its tools are a handful of general theorems that hold in every such space. Today these are called Banach spaces, and the theory is called functional analysis. Quantum mechanics, partial differential equations, numerical analysis and probability all use it as basic language.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order is roughly chronological: the thesis (1920), measure and paradox (1923–1924), uniform boundedness (1927), Hahn–Banach and open mapping (1929), and the monograph of 1932. Banach worked in a city that was Polish Lwów in his lifetime and is Ukrainian Lviv today; this course uses the name of the period.

Timeline

Kraków, 1892–1910

Stefan Banach was born on 30 March 1892 in Kraków, then part of the Austro-Hungarian province of Galicia. His parents were not married. His father, Stefan Greczek, served in the army and later worked as a tax official; his mother, recorded as Katarzyna Banach, was a domestic servant. Both came from the Podhale highlands south of the city. The boy took his father’s first name and his mother’s surname. After his baptism he was taken to his grandmother in Ostrowsko and then placed with a foster mother in Kraków, Franciszka Płowa, who raised him with her relative Maria. A French intellectual connected to the household, Juliusz Mien, taught him French, the language in which he later published most of his work.

In 1902 he entered the IV Gymnasium in Kraków. He was strong in mathematics, less so elsewhere, and passed the final school examination (the matura) in 1910 without distinction. One of his classmates, Witold Wilkosz, also became a professor of mathematics. Both young men reportedly believed that little new remained to be done in mathematics, and Banach decided to study engineering.

Engineering, war and a park bench, 1910–1920

In 1910 Banach moved to Lwów, the capital of Galicia, and enrolled in engineering at the Polytechnic. He supported himself by tutoring, and his studies stretched out longer than usual. When war broke out in 1914 he returned to Kraków. Poor eyesight kept him out of the army; he worked on road building, taught in local schools and attended some mathematics lectures at the Jagiellonian University. He had no formal training in mathematics, but he read widely on his own.

The Planty

In 1916 the mathematician Hugo Steinhaus, walking in the Planty, the park ring around Kraków’s old town, heard the words “Lebesgue integral” from a bench. The two young men there were Banach and Otto Nikodym. Steinhaus joined the conversation and told them about a problem on the convergence of Fourier series he had been unable to solve. A few days later Banach brought him the solution. Their joint paper appeared in 1918 in the Bulletin of the Kraków Academy, Banach’s first publication. Steinhaus later called Banach his greatest mathematical discovery. In 2016 the city placed a bench in the Planty with figures of Banach and Nikodym.

Through Steinhaus, Banach also met Łucja Braus, whom he married in 1920. In 1919 the mathematicians of Kraków founded a mathematical society, which in 1920 became the Polish Mathematical Society.

The thesis and the chair, 1920–1924

In 1920 Antoni Łomnicki offered Banach a post as his assistant at the Lwów Polytechnic, where Steinhaus was now also working. Banach had no university degree, but the University of Lwów accepted his doctoral dissertation as an exception. Its title, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, understates its content. In it Banach set out axioms for a complete normed vector space, proved general theorems about such spaces, and applied them to integral equations. It was published in 1922 in the third volume of Fundamenta Mathematicae. The Mathematics Genealogy Project lists Steinhaus and the philosopher Kazimierz Twardowski as his advisors.

Banach was not the only one to have the idea. Maurice Fréchet had introduced abstract metric spaces in 1906, and between 1920 and 1922 Hans Hahn, Eduard Helly and Norbert Wiener each wrote down some version of a normed space. Banach’s treatment was the most complete, and he and his school drew far more from it.

His rise was rapid. He completed his habilitation in 1922, and on 22 July 1922 was appointed professor extraordinary at the Jan Kazimierz University in Lwów. In 1924 he became full professor and a corresponding member of the Polish Academy of Learning, and spent the academic year 1924–25 in Paris. In these same years he wrote two papers that still surprise every reader: the construction of invariant measures on the line and plane (1923), and, with Alfred Tarski, the paradoxical decomposition of the ball (1924).

Complete normed spaces

For the general reader. Functions can be added and scaled, just like arrows in space. If we also have a way to measure the size of a function, we can talk about functions being close, about limits, and about continuity of operations on functions. Banach’s thesis listed the few rules such a measure of size must obey, and showed that a large part of analysis follows from those rules alone.

Examples

• \(C[a,b]\) with \(\|f\|_\infty=\max|f|\). Completeness is the theorem that a uniform limit of continuous functions is continuous.
• \(\ell^p\) (\(1\le p<\infty\)), sequences with \(\|x\|_p=(\sum|x_n|^p)^{1/p}<\infty\); \(\ell^\infty\) with the supremum norm; \(c_0\), sequences tending to 0.
• \(L^p(\mu)\), with completeness given by the Riesz–Fischer theorem.
• Every Hilbert space, with \(\|x\|=\sqrt{\langle x,x\rangle}\).

The triangle inequality for \(\|\cdot\|_p\) is Minkowski’s inequality. Geometrically, a norm is determined by its unit ball, which must be convex, balanced and absorbing; the hero illustration shows the balls of \(\ell^p\) in the plane.

Bounded operators

The contraction principle

For the general reader. Crumple a map of a city and drop it on the ground in that city. Exactly one point on the map lies directly above the place it represents. Banach’s thesis contains a sharp version of this fact: a map that shrinks all distances has exactly one fixed point, and repeating the map finds it. It is the simplest existence theorem in analysis and one of the most used.

Proof. By induction \(d(x_{n+1},x_n)\le q^nd(x_1,x_0)\). For \(m>n\) the triangle inequality gives \(d(x_m,x_n)\le(q^n+\cdots+q^{m-1})d(x_1,x_0)\le\frac{q^n}{1-q}d(x_1,x_0)\), so \((x_n)\) is Cauchy and converges to some \(x^*\). Continuity of \(T\) gives \(Tx^*=x^*\), and letting \(m\to\infty\) gives the estimate. If \(Ty=y\) too, \(d(x^*,y)\le q\,d(x^*,y)\) forces \(y=x^*\).

Integral equations

Banach’s motivating application was to Fredholm and Volterra equations. For \(f\in C[a,b]\), a continuous kernel \(K\) and a parameter \(\lambda\), consider

The right side defines \(T\) on \(C[a,b]\) with \(\|Tx-Ty\|_\infty\le|\lambda|\,(b-a)\max|K|\,\|x-y\|_\infty\). If \(|\lambda|(b-a)\max|K|<1\), there is a unique continuous solution, and the iterates converge to it. In operator language this is the Neumann series: if \(\|A\|<1\) in a Banach algebra of operators, then \(I-A\) is invertible and \((I-A)^{-1}=\sum_{n\ge0}A^n\).

The Hahn–Banach theorem and duality

For the general reader. In ordinary space, a convex body and a point outside it can be separated by a plane. Banach proved that this remains true in infinite-dimensional spaces of functions. The consequence is that every such space has many linear measurements, called functionals, enough to tell any two elements apart.

The dual space \(X^*\) is the space of bounded linear functionals on \(X\), itself a Banach space under \(\|\varphi\|=\sup_{\|x\|\le1}|\varphi(x)|\). A function \(p:X\to\mathbb R\) is sublinear if \(p(x+y)\le p(x)+p(y)\) and \(p(tx)=tp(x)\) for \(t\ge0\).

Hans Hahn proved the normed-space version in 1927; Banach, independently, proved the version above with a sublinear function in 1929, in the first volume of Studia Mathematica. The complex case followed in 1938 (Bohnenblust and Sobczyk; Sukhomlinov).

Proof sketch

One step. Take \(x_0\notin M\). We need a value \(c=F(x_0)\) with \(f(m)+tc\le p(m+tx_0)\) for all \(m\in M\), \(t\in\mathbb R\). Dividing by \(|t|\), this means

Such \(c\) exists because \(f(m)+f(m')=f(m+m')\le p(m+m')\le p(m-x_0)+p(m'+x_0)\). All steps. Order the extensions dominated by \(p\) by inclusion. Every chain has an upper bound (its union), so Zorn’s lemma gives a maximal extension, which by the one-step argument is defined on all of \(X\). Banach used transfinite induction, the form of choice available to him.

Baire category: three great theorems

For the general reader. In a complete space, a countable union of “thin” sets can never fill everything. This simple-looking fact, proved by René Baire for the real line, became in Banach’s hands a machine for theorems. It shows that pointwise good behaviour of a family of operators forces uniform good behaviour, and that the typical continuous function has no derivative anywhere.

Proof. The closed sets \(E_n=\{x:\sup_\alpha\|T_\alpha x\|\le n\}\) cover \(X\). By Baire one of them contains a ball \(B(x_0,r)\). For \(\|x\|\le r\), \(\|T_\alpha x\|\le\|T_\alpha(x_0+x)\|+\|T_\alpha x_0\|\le2n\), so \(\|T_\alpha\|\le2n/r\) for all \(\alpha\).

Juliusz Schauder gave a simplified proof of the open mapping theorem in 1930. Both theorems and the closed graph theorem appear in Banach’s 1932 book.

Most functions are nowhere differentiable

Karl Weierstrass had shocked the 1870s with a single continuous, nowhere differentiable function. In 1931 Banach and, independently, Stefan Mazurkiewicz showed that such functions are the rule.

Measure, paradox and the Banach–Tarski theorem

For the general reader. Can every set of points be given a size, so that sizes add up and do not change under rigid motions? On the line and in the plane, Banach showed in 1923 that the answer is yes, if one asks only for finite additivity. In space the answer is no, spectacularly: with Alfred Tarski he proved in 1924 that a solid ball can be cut into finitely many pieces and reassembled into two balls of the same size.

The measure problem

Giuseppe Vitali showed in 1905 that no countably additive, translation-invariant measure on all subsets of \(\mathbb R\) extends length. Felix Hausdorff showed in 1914 that in \(\mathbb R^3\) even finite additivity fails: the sphere admits a paradoxical decomposition up to a countable set.

The proof uses an extension argument of Hahn–Banach type. What makes dimensions 1 and 2 different is that their isometry groups are solvable, and solvable groups are amenable: they carry invariant means. This was the start of the theory of amenable groups, named by Mahlon Day after John von Neumann’s 1929 analysis of the paradox.

The paradox

Two sets \(A,B\subseteq\mathbb R^n\) are equidecomposable if \(A=A_1\sqcup\cdots\sqcup A_k\) and \(B=B_1\sqcup\cdots\sqcup B_k\) with \(B_i=g_iA_i\) for isometries \(g_i\).

Outline. (1) The rotations \(a,b\) by \(\arccos\tfrac13\) about two perpendicular axes generate a free group \(F_2\subset SO(3)\). (2) \(F_2\) is paradoxical: writing \(W(x)\) for reduced words beginning with \(x\), \(F_2=\{e\}\sqcup W(a)\sqcup W(a^{-1})\sqcup W(b)\sqcup W(b^{-1})\), and \(F_2=W(a)\sqcup aW(a^{-1})=W(b)\sqcup bW(b^{-1})\). (3) \(F_2\) acts freely on the sphere minus the countable set \(D\) of fixed points; choosing one point from each orbit (axiom of choice) transfers the paradox to \(S^2\setminus D\). (4) \(D\) is absorbed by a rotation, and radial extension plus the centre handle the ball. (5) The Banach–Schröder–Bernstein theorem, proved in the same paper, lets one pass from “\(A\) is equidecomposable with a subset of \(B\) and vice versa” to “\(A\) and \(B\) are equidecomposable.” Raphael Robinson showed in 1947 that five pieces suffice for the ball, and no fewer.

The pieces cannot all be Lebesgue measurable, since measure would then be preserved. The paradox therefore marks the boundary between measurable geometry and what the axiom of choice permits.

Weak topologies, bases and the goose

For the general reader. In infinite dimensions, bounded sequences need not have convergent subsequences. Banach recovered compactness by weakening what convergence means. He also asked whether every space of functions has a “coordinate system” like the standard basis of \(\mathbb R^n\). That question waited forty years for its answer, and its solver received a live goose.

A sequence \(x_n\) converges weakly to \(x\) if \(\varphi(x_n)\to\varphi(x)\) for every \(\varphi\in X^*\). Functionals \(\varphi_n\in X^*\) converge weak* to \(\varphi\) if \(\varphi_n(x)\to\varphi(x)\) for every \(x\in X\).

Banach proved the sequential version for separable spaces in his book; Leonidas Alaoglu proved the general version in 1940 using Tychonoff’s theorem.

Bases

A Schauder basis of \(X\) (Schauder, 1927) is a sequence \((e_n)\) such that every \(x\in X\) has a unique expansion \(x=\sum_n a_ne_n\) converging in norm. The unit vectors form a basis of \(\ell^p\) (\(p<\infty\)) and \(c_0\). Schauder constructed a basis of \(C[0,1]\) from piecewise linear “tent” functions.

Banach’s book asked whether every separable Banach space has a basis. Mazur, in Problem 153 of the Scottish Book (1936), asked an equivalent form of the related approximation question, and offered a live goose for its solution.

Per Enflo announced the result in 1972 and received the goose from Mazur in Warsaw. Other results of the 1932 book include the theorem of Banach and Mazur that every separable Banach space is isometric to a closed subspace of \(C[0,1]\), a theory of linear dimension comparing spaces by isomorphic embeddings, and the Banach–Mazur distance \(d(X,Y)=\inf\{\|T\|\,\|T^{-1}\|\}\) between isomorphic spaces.

Real functions: the indicatrix and the game

For the general reader. Before and alongside functional analysis, Banach worked on the classical theory of real functions. Two of his ideas from that world are still in use. One counts how many times a graph crosses each horizontal line and adds the counts up to get the total up-and-down travel of the function. The other is a game, posed by Mazur in the Scottish Book, in which two players take turns shrinking intervals, and which turned out to measure exactly how large a set is in Baire’s sense.

The indicatrix

For a continuous \(f:[a,b]\to\mathbb R\), let \(N_f(y)\in\{0,1,2,\dots,\infty\}\) be the number of solutions of \(f(x)=y\). This is the Banach indicatrix of \(f\).

Proof idea. Divide \([a,b]\) into \(2^n\) equal intervals \(I_{n,k}\) and let \(\chi_{n,k}\) be the indicator of \(f(I_{n,k})\), an interval of length \(\operatorname{osc}_{I_{n,k}}f\). Then \(N_n=\sum_k\chi_{n,k}\) increases to \(N_f\) (up to the countably many values at division points), and \(\int N_n=\sum_k\operatorname{osc}_{I_{n,k}}f\) increases to \(V_a^b(f)\). Monotone convergence gives the equality. The formula is the one-dimensional case of the area formula of geometric measure theory.

The Banach–Mazur game

Fix a set \(A\subseteq[0,1]\). Player I chooses a closed interval \(J_1\); player II chooses a closed interval \(J_2\subseteq J_1\); player I chooses \(J_3\subseteq J_2\), and so on. Player II wins if \(\bigcap_nJ_n\subseteq A\); otherwise player I wins. Mazur posed the game as Problem 43 of the Scottish Book and conjectured the answer below; Banach proved it, though his proof was not published. John Oxtoby gave the first published proof and a general form in 1957.

The Lwów school and the Scottish Café, 1924–1939

By the late 1920s Lwów had a mathematical school to match Warsaw’s. Warsaw, with Wacław Sierpiński, Kazimierz Kuratowski and Fundamenta Mathematicae, concentrated on set theory and topology. Lwów, around Banach and Steinhaus, concentrated on functional analysis. In 1929 the two founded Studia Mathematica, devoted mainly to that subject, and in 1931 Banach launched the series Monografie Matematyczne. Its first volume, Théorie des opérations linéaires (1932), a French version of his Polish text of 1931, was the first monograph on functional analysis. It made the field and its notation international.

Around Banach gathered Stanisław Mazur, his doctoral student (1932), and Władysław Orlicz, Juliusz Schauder, Stanisław Ulam, Józef Schreier, Herman Auerbach, Mark Kac and others. Kuratowski taught at the Lwów Polytechnic from 1927 to 1934 and wrote papers with Banach. The style of work was collective and informal. Much of it happened in cafés, first the Roma and then the Scottish Café, where discussions could last many hours.

The Scottish Book

To keep problems from being lost on the marble tabletops, Łucja Banach is credited with buying a thick notebook, which the café kept and produced on request. The Scottish Book collected 193 problems, many with prizes attached, from a small beer to a bottle of brandy to Mazur’s goose. Banach wrote 14 problems alone and many more with Mazur and Ulam. The last entry, by Steinhaus, is dated 31 May 1941. Ulam, who had emigrated to the United States in the 1930s, circulated an English translation from Los Alamos in 1957, and an edited edition with commentary appeared in 1981.

Banach gave a plenary address at the International Congress of Mathematicians in Oslo in 1936, and in 1939 he was elected president of the Polish Mathematical Society. He also wrote textbooks, including school arithmetic and university courses in calculus and mechanics.

Occupation, death and legacy, 1939–1945

In September 1939 the Soviet Union occupied Lwów under the Molotov–Ribbentrop pact. The university was reorganized as a Ukrainian institution named after Ivan Franko. Banach kept his chair, had to undertake to learn Ukrainian, and became dean of the Faculty of Science. He was elected a corresponding member of the Ukrainian Academy of Sciences, and sat on the city council from 1940.

In late June 1941 German troops took the city. The universities were closed. Within days the occupiers murdered dozens of Polish professors, among them Łomnicki. Schauder, Auerbach and many other Jewish members of the school were killed during the occupation. From late 1941 to July 1944 Banach survived as a lice feeder at the typhus research institute of the biologist Rudolf Weigl, who used the work to shelter many scholars. Feeders let infected lice feed on their skin, in return for papers that gave some protection from arrest and deportation.

After the Red Army returned in July 1944, Banach helped reopen the university. He was offered a chair at the Jagiellonian University in Kraków and was mentioned as a possible Polish minister of education. In January 1945 he was diagnosed with lung cancer. He died in Lwów on 31 August 1945, aged 53, and was buried in the Lychakiv Cemetery after a funeral attended by hundreds.

Legacy

• Mathematics: Banach spaces are the setting for linear analysis. The theorems of Lessons 05–07 are taught in every graduate course. Banach algebras are the basis of Israel Gelfand’s spectral theory and of operator algebras.
• Geometry of Banach spaces: the questions of his book, on bases, isomorphism types and embeddings, grew into a large field. Many were settled only after 1970 by Enflo, by Timothy Gowers and Bernard Maurey, and others.
• Set theory and groups: the Banach–Tarski paradox led to the theory of amenable groups and remains the standard illustration of the axiom of choice.
• Honours: the Polish Mathematical Society’s Stefan Banach Prize (from 1946), the Banach Medal of the Polish Academy of Sciences (from 1992), the International Stefan Banach Prize (from 2009), the Stefan Banach International Mathematical Center in Warsaw, and the minor planet 16856 Banach.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Stefan Banach, MacTutor History of Mathematics.
2. Stefan Banach, Wikipedia.
3. Stefan Banach, Mathematics Genealogy Project.
4. Scottish Book, Wikipedia.
5. R. Kałuża, Through a Reporter’s Eyes: The Life of Stefan Banach, Birkhäuser, 1996.
6. E. Jakimowicz and A. Miranowicz (eds.), Stefan Banach: Remarkable Life, Brilliant Mathematics, Gdańsk University Press, 3rd ed. 2011.
7. S. M. Ulam, Adventures of a Mathematician, Scribner, 1976.

Mathematics

1. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181.
2. S. Banach, Sur le problème de la mesure, Fund. Math. 4 (1923), 7–33.
3. S. Banach and A. Tarski, Sur la décomposition des ensembles de points en parties respectivement congruentes, Fund. Math. 6 (1924), 244–277. Volume contents.
4. S. Banach and H. Steinhaus, Sur le principe de la condensation de singularités, Fund. Math. 9 (1927), 50–61.
5. S. Banach, Sur les fonctionnelles linéaires I, II, Studia Math. 1 (1929).
6. S. Banach, Théorie des opérations linéaires, Monografie Matematyczne 1, Warsaw, 1932. English translation: Theory of Linear Operations, North-Holland 1987, Dover 2009.
7. R. D. Mauldin (ed.), The Scottish Book: Mathematics from the Scottish Café, Birkhäuser, 1981; 2nd ed. 2015.
8. G. Tomkowicz and S. Wagon, The Banach–Tarski Paradox, 2nd ed., Cambridge University Press, 2016.
9. P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317.
10. W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, 1991.
11. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.