Biorthogonal system

Ok kiddo, let me explain what a biorthogonal system is.

Imagine you have a bunch of toys, like blocks, balls, and dolls. We can arrange them in different ways, like building a tower with the blocks or throwing the balls to different directions.

In mathematics, we can also arrange some things in different ways. We can have a set of functions, like f(x) and g(x), and we can use them to describe different things, like the temperature or the speed of a car.

A biorthogonal system is like having two sets of toys that can be arranged in a special way. It means that we have two sets of functions that are related to each other in a specific way. Let's call them {f(x)} and {g(x)}.

The interesting thing is that when we multiply f(x) by g(x) and add up all the results, it gives us a special number (called the inner product) that tells us how similar these two sets of functions are.

But here's the tricky part: we need to make sure that if we multiply f(x) by any other function from the {g(x)} set and add up all the results, it gives us zero (or very close to it). And the same thing should happen if we multiply g(x) by any other function from the {f(x)} set.

That means that these two sets of functions are kind of like friends. They are related in a special way that makes them a good team. And we can use this special relationship to do many cool things in math, like solving equations or understanding patterns better.

So, in summary, a biorthogonal system is like having two sets of functions that are related to each other in a special way, and when we multiply them together and add up the results, it gives us some important information about how these two sets of functions are similar to each other.