The Polish Virtual Library of Science (which contains Banach’s book) has several other interesting mathematical works in English, French, German, and Polish. They have the archives for several journals, including Studia Mathematica. They also have several older but famous monographs, such as Zygmund’s Trigonometric Series.

Stefan Banach’s famous monograph, Théorie des opérations linéaires, which gave birth to the field of Banach spaces, is available online.

I’ve been doing some more online reading on the Positivstellensatz. I had blithely assumed that the polynomials that are non-negative for all real values were given by sums of squares of polynomials, but this is false. What is true is that a polynomial is non-negative if and only if it can be written as the sums of squares of rational functions, but this is a nontrivial result. In fact, showing this was Hilbert’s seventeenth problem. (You can derive it from the Positivstellensatz.)

If you were to try to axiomatize the idea of positivity inside a commutative ring in such a way that the same set of axioms cover both positive real numbers and non-negative polynomials, you would include axioms such as the sum and product of two positive elements are positive, and that squares are always positive. Hilbert’s seventeenth problem shows that you need an additional axiom: if a and ab are positive, then so is b.

In algebraic geometry, the Nullstellensatz gives an algebraic characterization of when a multivariate polynomial vanishes on a set of points defined by a system of (complex) polynomial equations.

I’ve just come across an analogue of this over the reals. In this case, it’s a purely algebraic characterization of when a multivariate polynomial is guaranteed to be positive over a set of points defined by a system of polynomial inequalities. In analogy with the complex case, the result is known as the Positivstellensatz.