We present a new proof of the Pythagorean theorem which suggests a particular decomposition of the elements of a topological algebra in terms of an “inverse norm” (addressing unital algebraic structure rather than simply vector space structure). One consequence is the unification of Euclidean norm, Minkowski norm, geometric mean, and determinant, as expressions of this entity in the context of different algebras.

Apart from being unital topological *-algebras, matrix algebras, special Jordan algebras and Cayley-Dickson algebras would seem to have little else in common. For example, the matrix algebra

on an

on the algebra

on the matrix algebra

on the spin factor Jordan algebras,

There is another aspect relevant to the Cayley-Dickson algebras. In addition to (

So, we first prove the Pythagorean Theorem by deriving (

It will also be seen that there is a hierarchy related to the basic equations, evidenced by progressively more structure accompanying the solution function on particular algebras. The equations (

In a space satisfying the axioms of plane Euclidean geometry, the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two other sides.

The theorem hypothesis is assumed to indicate the Hilbert formulation of plane Euclidean geometry [

A vector space structure is defined on the Euclidean plane

We now fill in details the above argument.

Given a right triangle

A

For any point

Let the length of

Thus, suppose

Let

We are first required to show the existence of the limit in (

We have

If

Thus, suppose

Next, we establish the continuity of each directional derivative. Consider any sequence of line segments

We next show that the largest directional derivative at

Now,

Diagram relating to proof that the function giving the length of a line segment with one endpoint at

Definition

The multiplication of a scalar with a point,

The “sum” of two points,

If

If either

If

If

when

when

if

if

if

With this understanding, it is clear that

The central role of the derivative of the norm function as featured in Definition

There is an isomorphism between the vector space

We first identify a particular Cartesian axis system on the plane. One Cartesian system is already present, consisting of the lines containing the line segments forming the right angle of the right triangle given at the outset of this proof (i.e.,

We identify

The above is therefore a one-to-one mapping between the points of the Euclidean plane and the points of

It follows that the isomorphism in the above lemma leads to an expression for the directional derivative in the vector space

In fact, it is easy to see that the equations of the last sentence of the prior paragraph pertain not just

Equations (

A function

Equations (

But the Euclidean norm does not address the multiplicative structure of an algebra and so does not have an essential role in most algebras. Instead, we shall see that the role of

The “defining” equations of the Euclidean norm, (

For a topological

The above equations mimic the equations for the Euclidean norm referred to in Corollary

Of course, from (

The Euclidean norm is an inverse norm on the Cayley-Dickson algebras.

However, inverse norms have applicability well beyond the Cayley-Dickson algebras.

For

The above is a well-known immediate consequence of the Jacobi’s formula in matrix calculus (the latter expresses gradient of the determinant in terms of the adjugate matrix).

For the algebra of real matrices

In analogy with Euclidean decomposition (

For the algebra

Now we turn to Jordan algebras.

The Minkowski Norm is an inverse norm on the spin factor jordan algebra.

Thinking of

Now we define

Applying

On the other hand, the Jordan algebra obtained from the algebra of matrices

Supplying an inverse norm nominally requires solution of a nonlinear partial differential equation (

However, not all unital algebras have an inverse norm satisfying (

The unital hull of a nonunital topological algebra does not have an inverse norm satisfying (

The unital hull of a nonunital algebra

One can get even more restrictive and consider algebras for which not only is