Inner product spaces

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 and , such that $Multiple \tag(x, y)=\overline{(y, x)}, \tag{1}\]\[ (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 is always real, so that the inequality in (3) makes sense. In an inner product space we shall use the notation the number is called the norm or length of the vector . 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 and ; in the first case we write, for and , and, in , we write The modifications that convert these examples into Euclidean spaces (that is, real inner product spaces) are obvious.

In a unitary space we have " transform="translate(0,779.5)">(To transform the left side of (2 ) into the right side, use (1), expand by (2), and use (1) again.) This fact, together with the definition of an inner product, explains the terminology sometimes used to describe properties (1), (2), (3) (and their consequence (2 )). According to that terminology is a Hermitian symmetric (1), conjugate bilinear ((2) and (2 )), and positive definite (3) form. In a Euclidean space the conjugation in (2 ) may be ignored along with the conjugation in (1); in that case is called a symmetric, bilinear, and positive definite form. We observe that in either case, the conditions on imply for the homogeneity property (Proof: .)

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