0

Vector spaces may not be equipped with an inner product (for instance this question). Mathematically, one can study such spaces in its own right, but I was just wondering: is there a scientific application to such spaces? For instance I will try to imagine a physical system that can be described with a vector space without an inner product but I can't come up with an easy example.

My guess is that maybe for some stochastic equations describing a system this could be achieved, but I'm not sure. I appreciate if example in Physics could be provided, but other fields (e.g. Computer Science) are welcome.

user2820579
  • 2,389

1 Answers1

3

As you point out, finite fields may not be equipped with an inner product. But one may have vector spaces over such finite fields. See here for more answers: Explicit example of a vector space over a finite field, and linear transformation of vector spaces over different fields

For a practical application, consider some sequence of Random Variables $\{X_t\}_{t=1}^\infty$ that may take only discrete values ($X\in K$) where $K$ is a some finite field (or cartesian product of finite fields). Then one may want to model evolution of X over such a field, for which the vector space formulation may be useful.

You may want to look up elliptic curve cryptography for some concrete examples where finite fields are used in CS.

Rahul Madhavan
  • 2,789
  • 1
  • 11
  • 14