Quote:
Originally Posted by AntonioLao  To exist, a triangle plane geometric figure with sides a, b, and c must satisfy the following inequalities: 1. a+b>c, 2. a+c>b, 3. b+c>a. Any two sides can be equal but their sum cannot be less than the third. When all sides are equal then it is an equilateral triangle representing a plane figure of perfect symmetry. The export of these existence conditions by particular definitions becomes the import of a generalized theorem of abstract finite-dimensional spacetime structures. These have been extended into infinite dimensional and also defined over the field of complex numbers with inner products, for example: Hilbert spacetime structures in quantum mechanics. However, these are inherently one-dimensional. Higher dimensions would require outer product spacetime descriptions. Their abstraction still does not exist in the annuals of mathematics.
A real finite-dimensional vector space on which an inner product is defined is usually called a Euclidean space. On the other hand, if extended over the field of complex numbers, it is called a unitary space. If X represents an n-dimensional Euclidean space over the field of real numbers, R, then a mapping h of R into R is called a real bilinear form described by matrices or linear transformations. When and where used to represent space-time structures for squares of energy, they are singular square Hadamard matrices satisfying Schwarz inequality given by |(x,y)|≤|x|∙|y|, where x and y are Hadamard matrices of arbitrary dimensions. |
I somehow feel there is a fusion link lurking nearby!
regards michael.