Elementary Functional Analysis

Abstract

Elementary functional analysis, such as Hilbert spaces, Sobolev spaces, and linear operators, is collected in this chapter for the reader’s convenience. Since elementary stabilization results are presented without using functional analysis, readers may skip this chapter and come back when needed. In this book, ℕ denotes the set of all nonnegative natural numbers, ℂ denotes the set of complex numbers, ℝⁿ denotes the n-dimensional Euclidean space, and ℝ = ℝ¹ denotes the real line. Points in ℝⁿ will be denoted by x = (x ₁, …, x _n ), and its norm is defined by

$$\|\mathbf{x}\| = \left(\sum_{i = 1}^{n}x_{i}^{2}\right)^{\frac{1}{2}}.$$