The Importance of the Boundary

How is it possible to describe any analytic or harmonic function on a compact set in terms of much simpler “construction blocks” like polynomials? Or, how is it possible to know the values of a function inside a compact domain, by knowing only its values on the boundary* Well, these simplifications are possible because the “bricks” are actually organized in complicated and versatile structures. For example the B^* algebras. And in addition, the compact domains are certainly among the simples ones, being always reducible to finite reunions. Actually, a complicated structure like a B^* algebra, defined by 24 axioms (out of which 13 axioms on commutative algebras, five axioms on norm, one for completeness, and five more specific axioms) can be realized by continuous functions defined on a compact set. It is not the only example. The space l₁ of complex sequences with norm given by the sum of the modules of the terms is isomorphic with the algebra of functions whose Fourier series is absolutely convergent. Also, a compact Hausdorff space, with topology induced by distance, is homeomorphic with a compact subset of [0, 1]^N. Any two separable Hilbert spaces are isomorphic, and so on. These similarities bring a unifying point of view: objects of apparently distinct nature, like Weierstrass-Stone theorem on function approximation, Wiener theorem on absolutely convergent Fourier series, spectral expansion of self-adjoint operators, the theorems of Tikhonov, Stone-Čech, or the fixed-point theorem of Brouwer, the Cauchy formula on complex functions, the Green representation theorem, the Poincaré Lemma, etc. actually provide the same fundamental truth: simplification by approximation is possible on compacts.