Hilbert Spaces

Abstract

Throughout this book, $\mathcal{H}$ is a real Hilbert space with scalar (or inner) product $\langle \cdot | \cdot \rangle$. The associated norm is denoted by $\parallel \cdot \parallel$ and the associated distance by d, i.e.,

$$(\forall x \in \mathcal{H}) (\forall y \in \mathcal{H}) \quad \parallel x \parallel = \sqrt {\langle x | x \rangle} \ {\rm and} \ d(x, y) = \parallel x-y \parallel.$$

(2.1)