Definition 1. An inner product in a (real or complex) vector space is a (respectively, real or complex) numerically valued function of the ordered pair of vectors \(x\) and \(y\) , such that \[(x, y)=\overline{(y, x)}, \tag{1}\]\[(\alpha_{1} x_{1}+\alpha_{2} x_{2}, y)=\alpha_{1}(x_{1}, y)+\alpha_{2}(x_{2}, y), \tag{2}\]\[ (x, x) \geq 0; \quad (x, x)=0 \text { if and only if } x=0. \tag{3}\] An inner product space is a vector space with an inner product.
We observe that in the case of a real vector space, the conjugation in (1) may be ignored. In any case, however, real or complex, (1) implies that \((x, x)\) is always real, so that the inequality in (3) makes sense. In an inner product space we shall use the notation \[\sqrt{(x, x)}=\|x\|;\] the number \(\|x\|\) is called the norm or length of the vector \(x\) . A real inner product space is sometimes called a Euclidean space ; its complex analogue is called a unitary space .
As examples of unitary spaces we may consider \(\mathbb{C}^{n}\) and \(\mathcal{P}\) ; in the first case we write, for \(x=(\xi_{1}, \ldots, \xi_{n})\) and \(y=(\eta_{1}, \ldots, \eta_{n})\) , \[(x, y)=\sum_{i=1}^{n} \xi_{i} \bar{\eta}_{i},\] and, in \(\mathcal{P}\) , we write \[(x, y)=\int_{0}^{1} x(t) \overline{y(t)}\,dt.\] The modifications that convert these examples into Euclidean spaces (that is, real inner product spaces) are obvious.
In a unitary space we have \begin{align} (x, \alpha_{1} y_{1}+\alpha_{2} y_{2})=\bar{\alpha}_{1}(x, y_{1})+\bar{\alpha}_{2}(x, y_{2}). \tag{2