You might know that when a real number 'x' satisfies

it is also true that

Amazingly, this can be thought of as a single case of a greater idea, the convergence of a Neumann series! Instead of a real number, Neumann series deal with operators. An operator is a function that turns one element of a vector space into another element of another vector space. The same thing is meant by this notation, if A is an operator that takes elements of V and spits out elements of W:

If A is 'defined on V' that means

If A is linear, that means

Where 'x' and 'y' are elements of V, and c is something called a scalar.

Previous knowledge of linear algebra may mean you know many examples of this in the form of matrices and linear transformations! For those who have no clue what that means, multiplication of some real number is a fine example of an operator. Check for yourself if 'multiplying a real number by 8' satisfies the properties of a linear operator defined on the set of real numbers.

represents the new operator resulting from applying the operator T, k times.

For example, if

then

A Neumann series is just like a power series. But now instead of adding exponents of real numbers, we're adding the repeated applications of an operator. If the operator is both linear and defined on a normed vector space, and the Neumann series of T converges (meaning that the limit of n to infinity of T^n is the 0 operator), consider the partial sums

Let's try going through the proof of the best friend identity, but now in the language of these more general operators. The 'Id' symbol you see here is called the identity operator. Id(x)=x.

This means that (Id-T) is the inverse of the limit of the partial sum, which is our Neumann series! So we finally have our best friend, but stronger!

ULTIMATE FORM BEST FRIEND!!!!

WOW, this is way too cool! I have not seen this before.