Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals.
Suppose the two linear functionals L1 and L2 are the same, say L. Then AF becomes a commutative algebra with multiplication defined by X is a nonempty set and F a field.
FX is the set of functions from X to F. If f, g are in FX, x in X and α in F, then define and With addition and scalar multiplication defined as this, FX is a vector space over F. Now, fix two elements a, b in X and define a function e from X to F by e(x) = 1F for all x in X.
Note that Let C denote the field of Complex numbers.
A continuous function γ from the closed interval [0, 1] of real numbers to the field C is called a curve.
The set V[0, 1] of all the curves is a vector space over C. We can make this vector space of curves into an algebra by defining multiplication as above.
we have for α,β in C[0, 1], Then, V[0, 1] is a non-commutative algebra with e as the unity.
Let us take (1) the line segment joining the points (1, 0) and (0, 1) and (2) the unit circle with center at the origin.
Book of Beautiful Curves Certain Number Theoretic Episodes in Algebra