What This Book Is About: Approximants

The key concept of this book is that of an approximant approximant (the characteristically snappy term is due to Halmos [21]). Let 𝕃 $$\mathbb{L}$$ , say, be a space of mathematical objects (complex numbers or square matrices, say); let ℕ $$\mathbb{N}$$ be a subset of 𝕃 $$\mathbb{L}$$ each of whose elements have some “nice” property p (of being real or being self-adjoint, say); and let A be some given, not nice element of 𝕃 $$\mathbb{L}$$ ; then a p-approximant of A is a nice mathematical object that is nearest, with respect to some norm, to A. In the first example just mentioned, a given complex number z has its real part ℝ z ( = z + z ̄ 2 ) $$\mathbb{R}z(={ z+\bar{z} \over 2} )$$ as its (unique) real approximant. In the second example, a given square matrix A has (by Theorem 3.2.1 ) its real part ℝ A ( = A + A ∗ 2 ) $$\mathbb{R}A(={ A+A^{{\ast}} \over 2} )$$ as its unique self-adjoint approximant.