Preface – I'm very new to Abstract Algebra, so if my terminology/interpretations are off, my apologies.
So I was thinking about real valued continuous non zero functions (from here I'll just refer to them as functions) and that when they are periodic of the same period (from here Ill just say periodic) we observe that the product of two periodic functions is periodic as is the sum of two periodic functions. After some easy work, I found that under the real definition of addition and multiplication that they form a Field with the Zero Element $0(x) = 0$ and the One Element $1(x) = 1$ and additive and multiplicative inverses defined as for real values, i.e. $-a(x) = -1 \times a(x)$ and $a(x)^{-1} = \frac{1}{a(x)}$
Anyways, it got me thinking about how structures that have periodic and non periodic functions could be defined. I may be wrong, but given that the sum of two periodic functions is periodic and the product of a periodic with a non periodic is periodic that we can define a Vector Space $(V,+,\times)$ over the Field $F$ where
$V$ = set of periodic functions
$F$ = set of all non-zero functions
Vector addition is the real addition of two periodic functions. Scalar Multiplication is the real multiplication of a non-zero function with a periodic function.
I believe this may be OK, but I'm concerned that the Scalar Field contains Vectors – i.e. a periodic function is a function and thus in F. I suppose I could circumnavigate this by defining F as non periodic non-zero functions but I'm not sure if I'm allowed to have it as it currently is defined.
Are there any pointers/references people can suggest.
Thanks,
David
Ps – Another form of this would be if we took integers divisible by a given integer k (addition of two divisible by k values and scalar mult of any integer with one divisible by k).
Edit _ as pointed out if can’t occur as the function set do not form a field.
Best Answer
You can't use the set of functions as scalars since they don't form a field. Functions that have zeroes somewhere don't have inverses. This has nothing to do with periodicity.
Restricting the scalars to functions that are never $0$ won't fix things. The sum of two such functions can have a root, so the set of those functions aren't closed under addition.
Your last PS is confusing, but hints at using the field of integers modulo a prime $p$ for the scalars. That will work: see the wikipedia entry on finite fields
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