On the Types of Functions which can serve as Scalar Products in a Complex Linear Space
Document Type
Article
Publication Date
1973
Department
Department of Mathematical Sciences
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.
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, 83-96.
http://doi.org/10.1016/0024-3795(73)90008-6
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/5390
Publisher's Statement
© 1973