Inner product

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scalar product, dot product, of two non-zero vectors and

The product of their lengths and the cosine of the angle between them:

is taken to be that angle between the vectors not exceeding . When or is zero, the inner product is taken to be zero. The inner product is called the scalar square of the vector (see Vector algebra).

The inner product of two -dimensional vectors and over the real numbers is given by

In the complex case it is given by

An infinite-dimensional vector space admitting an inner product and complete with respect to it is called a Hilbert space.


More generally, an inner product on a real vector space is a symmetric bilinear form which is positive definite, i.e., for all . A (unitary) inner product on a complex vector space is likewise defined as a Hermitian (i.e., with ) sesquilinear form, with complex conjugation as automorphism, which is positive definite. In finite-dimensional spaces one can always find an orthonormal basis in which takes the standard form , respectively .

Besides the inner product (which can be defined in arbitrary dimensions), in three-dimensional space one also has the vector product.


[a1] V.I. Istrăţescu, "Inner product structures" , Reidel (1987)
How to Cite This Entry:
Inner product. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article