Hilbert Spaces

Abstract

Our starting point is an infinite-dimensional complex vector space X. An inner product (,) in X is a mapping from $X \times X$ into $\mathbb{C}$ such that $(\alpha{x}+ \beta{y},Z) = \alpha(x,y)+ \beta(y,z),$ $(x,\alpha{y}+\beta{z})= \bar{\alpha}(x,y)+\bar{\beta}(x,z),$ $(x,x)\geq 0,$ $(x,x)=0\Leftrightarrow x=0$ for all x, y and z in X and all complex numbers $\alpha$ and $\beta$ Given an inner product (,) in X, the induced norm ║ ║ in X is given by

$$||x||^2 = (x,x),\quad x\epsilon X.$$