The $d$-dimensional Hilbert space ${\cal H} = R^d$

This vector space has a basis of unit vectors ${\bf e}_n$,

$${\bf e}_1 = \left( \begin{array}{l} 1\\ 0\\ ..\\ 0 \end{array}... ...uad , {\bf e}_d = \left( \begin{array}{l} 0\\ 0\\ ..\\ 1 \end{array}\right).$$ (168)

Vectors are columns and in printed text written as the transposed (symbolized by $^T$ that is sometimes omitted) of lines ${\bf x}=(x_1,...,x_d)^T$. The scalar product of two vectors ${\bf x}=(x_1,...,x_d)^T, {\bf y}= (y_1,...,y_d)^T$, is $\langle {\bf x} \vert {\bf y} \rangle = \sum_{n=1}^d x_n y_n$. Each vector ${\bf x}=(x_1,...x_d)^T \in R^d$ can be decomposed into

$${\bf x}=\sum_{n=1}^d\langle {\bf e}_n\vert{\bf x} \rangle{\bf e}_n.$$ (169)