#### Title

On the types of functions which can serve as scalar products in a complex linear space

#### Document Type

Article

#### Publication Date

1-1-1973

#### Abstract

In this paper a generalized inner product 〈x|y〉 is defined as a binary function with complex values which satisfies the following: (i) for any nonzero vector y and any complex number ζ there exists a vector x such that 〈x|y| = ζ, (ii) 〈x1 + x2|y1 + y2〉 = 〈x1|y1〉 + 〈x2|y1〉 + 〈x1|y2〉 + 〈x2|y2〉, (iii) 〈y|x〉 = f[〈x|y〉], where f is a continuous function and, (iv) 〈x|μy〉 = g[μ, 〈x|y〉], where g is a continuous function. These conditions induce several functional equations which are then solved. By making a linear combination of 〈x|y〉 with its complex conjugate a new function (x|y) is obtained which is either symmetric, antisymmetric, or Hermitian. The functions 〈x|y〉 and (x|y) have the same orthogonal vectors. © 1973.

#### Publication Title

Linear Algebra and Its Applications

#### Recommended Citation

Heuvers, K.
(1973).
On the types of functions which can serve as scalar products in a complex linear space.
*
Linear Algebra and Its Applications,
6*(C), 83-96.
http://doi.org/10.1016/0024-3795(73)90008-6

Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/5390