Riesz representation theorem

Riesz Representation Theorem is a big, fancy theorem that helps us turn some abstract math ideas into things we can actually see and measure in the real world.

Imagine you have a friend who loves to collect different types of fruits, like apples, bananas and oranges. You want to understand what your friend is collecting, but he doesn't actually tell you what he's collecting. Instead, he gives you a list of facts about his fruit collection. He tells you how many apples he has, how many bananas he has, and how many oranges he has. He even tells you how many of each type of fruit he has of a certain weight or size.

Now, imagine you didn't know what fruit was but knew these facts about your friend's collection. Riesz Representation Theorem says that you could still figure out what type of fruits your friend is collecting by understanding how the different types of fruits are related to each other based on their weights and sizes.

In more technical terms, Riesz Representation Theorem helps us understand how to turn abstract mathematical ideas, called linear functionals, into something we can measure in the real world. It says that any linear functional (a rule that takes in a mathematical object and spits out a real number) can be represented by an inner product with another mathematical object. This means that we can understand what a linear functional is doing by measuring the inner product of that functional with other mathematical objects.

Overall, Riesz Representation Theorem is a really powerful tool for helping us understand how abstract mathematical ideas can be connected to the real world, even when we don't fully understand those connections at first!