Analysis

Analysis studies limits, approximation, continuity, and infinite-dimensional spaces of functions and operators.

Operator theory is the largest field, with a quarter of the subject's articles. Its largest part, shift, Toeplitz and Hankel operators, starts from the unilateral shift on \(\ell^2(\mathbb{N})\), whose spectrum is the closed unit disc and whose Fredholm index is \(-1\). Concrete operators such as the Cesàro operator, bounded on \(\ell^2\) but not compact, sit beside general theory: the spectral theorem for compact self-adjoint operators, the homotopy invariance of the Fredholm index and the Courant–Fischer min–max theorem.

Complex analysis runs from Cauchy theory and the great Picard theorem through the Riemann mapping theorem to the Hardy, Bergman, Dirichlet and Fock spaces of analytic functions. Harmonic analysis covers Fourier series, with Fejér's theorem on the uniform convergence of the Cesàro means of continuous periodic functions, Fourier interpolation of radial Schwartz functions in dimensions \(8\) and \(24\), and discrete Fourier analysis on finite abelian groups and the Boolean cube.

Banach spaces are studied through duality and weak topologies, from the Hahn–Banach and Banach–Alaoglu theorems to James' reflexivity theorem, and through the classical sequence spaces \(\ell^p\) and \(c_0\). The rest of the subject includes special functions such as the gamma function and polylogarithms; real functions, from nowhere-differentiable functions to the Hyers–Ulam stability of additive maps; sequences and series, with Tauberian theorems and series for constants such as Catalan's constant; and numerical analysis, from Newton's method to Céa's lemma for Galerkin approximation.

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