Functional-theoretic algebra

Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals. In general, it is a non-commutative algebra. It becomes commutative when the two functionals are the same. 
Definition.
Let "AF" be a vector space over a field "F", and let "L"1 and "L"2 be two linear functionals on AF with the property "L"1("e") = "L"2("e") = 1"F" for some "e" in "AF". We define multiplication of two elements "x", "y" in "AF" by
It can be verified that the above multiplication is associative and that "e" is the identity of this multiplication. 
So, AF forms an associative algebra with unit "e" and is called a "functional theoretic algebra"(FTA).
Suppose the two linear functionals "L"1 and "L"2 are the same, say "L." Then "AF" becomes a commutative algebra with multiplication defined by
Example.
"X" is a nonempty set and "F" a field. "F""X" is the set of functions from "X" to "F".
If "f, g" are in "F""X", "x" in "X" and "α" in "F", then define 
and 
With addition and scalar multiplication defined as this, "F""X" 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") = 1"F" for all "x" in "X". 
Define "L"1 and "L2" from "F""X" to "F" by "L"1("f") = "f"("a") and "L"2("f") = "f"("b"). 
Then "L"1 and "L"2 are two linear functionals on "F""X" such that "L"1("e")= "L"2("e")= 1"F"
For "f, g" in "F""X" define 
Then "F""X" becomes a non-commutative function algebra with the function "e" as the identity of multiplication.
Note that 
FTA of Curves in the Complex Plane.
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 complex numbers "γ"(0) and "γ"(1) are, respectively,
the initial and terminal points of the curve. 
If they coincide, the
curve is called a "loop". 
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.
Choosing formula_7 we have for "α,β" in "C"[0, 1],
Then, "V"[0, 1] is a non-commutative algebra with "e" as the unity.
We illustrate
this with an example.
Example of f-Product of Curves.
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.
As curves in "V"[0, 1], their equations can be obtained as 
Since formula_10 the circle "g"
is a loop. 
The line segment "f" starts from :formula_11
and ends at formula_12 
Now, we get two "f"-products
formula_13 given by 
and
See the Figure.
Observe that formula_16 showing that 
multiplication is non-commutative. Also both the products starts from formula_17
References.
Book of Beautiful Curves
Certain Number Theoretic Episodes in Algebra