Exploring the Motivations of Compactness in Mathematics through the Riemann-Weierstrass Debate

This blog post delves into the origins of the concept of compactness in mathematics, based on the paper "Compactness and Dirichlet's Principle" by JK Seo and H. Zorgati (J. KSIAM, 2014). 

The concept of compactness was introduced in a pivotal debate between Bernhard Riemann (1826-1866) and Karl Weierstrass (1815-1897), which centered on the convergence issues within Dirichlet's principle concerning the minimization problem. Riemann's approach to Dirichlet's principle posited that for Dirichlet's problem, $\Delta u = 0$ in a  bounded smooth domain $\Omega \subset \mathbb{R}^3$ with boundary data $u|_{\partial\Omega} = \phi \in C(\partial\Omega)$, a solution $u$ could be derived as the limit of a minimizing sequence $\{u_n:n=1,2,\cdots\}$ for the energy functional $f(v) := \int_{\Omega} |\nabla v|^2 dx$, within the admissible set $\mathcal{A} := \{ v \in C^2(\Omega) \cap C(\overline{\Omega}) : v|_{\partial\Omega} = \phi\}$. Riemann essentially claimed  that this solution $u$ represents the critical point at which the energy functional $f$ attains its minimum within the admissible set.

Riemann's methodology was met with skepticism by the mathematical community due to its perceived lack of mathematical rigor. Weierstrass, in particular, critiqued Riemann's approach for its absence of a rigorous proof regarding the convergence of the sequence $\{u_n: n=1,2,\cdots\}$. This critique gained further momentum in 1870 when Weierstrass provided a compelling example of a one-dimensional minimization problem without existing minimizers. He examined the energy function $f(v)=\int_{-1}^{1} |x \frac{\partial}{\partial x} v(x)|^2 dx$ over the set $\mathcal{A} =\{ v \in C^1([-1,1]) : v(-1) = 0 , v(1) = 1\}$. In his example, the sequence defined by $u_n(x)=(\sin n\pi x/2)^2\chi_{[0,1/n]}(x)+ \chi_{(1/n,1]}(x)$  is a minimizing sequence because $\lim_{n\to \infty} f(u_n)= 0$.  However, the limit $\lim_{n\to \infty} u_n$ is  the Heaviside function $\chi_{(0,1]}(x)$, which does not belong to $\mathcal A$ because it is NOT continuous at zero. 

This notable critique is not a definitive counterexample against Riemann's employment of the Dirichlet principle, because it involves a degenerate PDE $\frac{d}{dx}(x^2 \frac{d}{dx} u)=0$ where the coefficient $x^2$ is zero at $x=0$, which violates the conditions stipulated by the Lax-Milgram theorem. This particular PDE contrasts sharply with the Laplace equation, which is characterized by its coercivity (a prerequisite to ensure the existence and uniqueness of a solution). It's important to note that in Riemann's time, the mathematical community had not fully developed the concepts of convergence. Moreover, foundational concepts such as compactness, measure theory, and Sobolev spaces had yet to be introduced. 

During the nineteenth century, the mathematical framework for quantifying the distance between functions was not yet fully developed, as concepts such as the $L^2$-norm and the $H^1$-norm had not been introduced. The supremum norm, defined as $\|u_n-u_m\|_\infty=\sup_{x\in\Omega}|u_n(x)-u_m(x)|$, is not appropriate for suitably representing the distance between two functions $u_n$ and $u_m$. In the late nineteenth century, during the formulation of the Ascoli-Arzela theorem, mathematicians employed the concepts of sequences of functions being uniformly equicontinuous and uniformly convergent. However, this concept is insufficient to derive a suitable convergence of the sequence $\{u_n: n=1,2,\cdots\}$.  

By conceptualizing the sequence of functions  $\{u_n: n=1,2,\cdots\}$ as points within an infinite-dimensional function space, the $L^2$-norm $\|u_n-u_m\|_2=\sqrt{\int |u_n -u_m|^2 dx}$ provides an analogy to the Euclidean distance between two points, applying the Pythagorean theorem. With the development of measure theory and  Sobolev spaces in the early to mid-20th century, it is possible to demonstrate that the limit $\lim_{n \to \infty} u_n$ exists in the $L^2$-sense, provided the condition $\sup_{n} \int |\nabla u_n|^2 dx < \infty$ is satisfied. This underscores the concept of compactness, illustrating that every bounded sequence in the Sobolev space $H^1(\Omega)$ possesses a convergent subsequence in $L^2(\Omega)$.

The motivation behind this blog post stems from my experience that many students pursuing studies in mathematic fields such as Analysis, Measure theory, and Topology are often unaware of the practical implications and foundational significance of topics. Typically, students encounter these subjects through highly abstract theories (such as the Heine-Borel theorem in Topology) or oversimplified examples, which may veil their relevance and practical applications. There's a critical demand for a unified teaching approach that not only deepens students' grasp of mathematical concepts but also enriches their appreciation by illuminating mathematics' vast relevance and its practical utility in real-world scenarios, well beyond academic boundaries.