Krein–Milman theorem
Okay kiddo, let me explain to you the Krein-Milman theorem. Imagine you have a big round cookie with lots of different chocolate chips on it. Each chocolate chip is a point on the cookie. Now, imagine you want to find the smallest shape that can contain all the chocolate chips on the cookie. That is what the Krein-Milman theorem does, but instead of chocolate chips it works with points in a mathematical space.
So, the Krein-Milman theorem says that if you have a closed and bounded set of points in a mathematical space (which means it has an outer edge and doesn't go on forever), then you can always find a smallest possible shape that contains all the points within it. This smallest shape is called the convex hull.
The convex hull is like drawing a line around all the points that stick out the furthest and then connecting those lines until you have a shape that includes all the points inside it. This might seem simple, but it's actually a pretty important theorem in mathematics because it can be used to prove other things, like the Hahn-Banach theorem.
In summary, the Krein-Milman theorem is like fitting a shape around a bunch of points in a mathematical space, just like finding the smallest shape that can contain all the chocolate chips on a big round cookie. The convex hull is the smallest possible shape that contains all the points, and this theorem is useful for proving other things in math.
So, the Krein-Milman theorem says that if you have a closed and bounded set of points in a mathematical space (which means it has an outer edge and doesn't go on forever), then you can always find a smallest possible shape that contains all the points within it. This smallest shape is called the convex hull.
The convex hull is like drawing a line around all the points that stick out the furthest and then connecting those lines until you have a shape that includes all the points inside it. This might seem simple, but it's actually a pretty important theorem in mathematics because it can be used to prove other things, like the Hahn-Banach theorem.
In summary, the Krein-Milman theorem is like fitting a shape around a bunch of points in a mathematical space, just like finding the smallest shape that can contain all the chocolate chips on a big round cookie. The convex hull is the smallest possible shape that contains all the points, and this theorem is useful for proving other things in math.