1. A scalar valued function of pairs of vectors from a vector space, denoted by (x,y ) where x and y are vectors, and with the properties that (x,x ) is always positive and is zero only if x = 0, that (ax + by,z ) = a(x,z ) + b(y,z ) for any scalars a and b, and that (x,y ) = (y,x ) if the scalars are real numbers, (x,y ) = (ȳ,x̄ ) if the scalars are complex numbers. Also known as Hermitian inner product; Hermitian scalar product.

2. The inner product of vectors (x₁, ..., x_n ) and (y₁, ..., y_n ) from n- dimensional euclidean space is the sum of x_iy_i as i ranges from 1 to n. Also known as dot product; scalar product.

3. The inner product of two functions f and g of a real or complex variable is the integral of f (x )ḡ(x̄ )dx, where ḡ(x̄ ) denotes the conjugate of g(x ).

4. The inner product of two tensors is the contracted tensor obtained from their product by means of pairing contravariant indices of one with covariant indices of the other.