Stone–Weierstrass theorem
Imagine you have a big bag of different shapes and sizes of stones, and you want to know if you can make any shape you want using only these stones. Stone–Weierstrass theorem says yes, you can do it!
So, how does it work? First, you need to know that there are a lot of different ways to measure the "shape" of a thing, such as its height or width. One way is to use a function, which is like a machine that takes in some numbers and gives you another number as the output.
The Stone–Weierstrass theorem says that if you have a function that you want to measure the shape of (let's call it f), and you have a bunch of simpler functions (let's call them g1, g2, g3, etc.), then you can combine these simpler functions in different ways to get really close to the shape of f.
Think of it like building with Legos. You have a bunch of different shapes and sizes, but you can put them together in different ways to build whatever you want. In the same way, you can use the simpler functions to build up the shape of f.
This is really useful because sometimes it's hard to work with complicated functions directly. But with the Stone–Weierstrass theorem, you can break them down into simpler pieces and work with those instead.
So, to sum it up: Stone–Weierstrass theorem says that you can build any function you want using simpler functions, kind of like building with Legos.
So, how does it work? First, you need to know that there are a lot of different ways to measure the "shape" of a thing, such as its height or width. One way is to use a function, which is like a machine that takes in some numbers and gives you another number as the output.
The Stone–Weierstrass theorem says that if you have a function that you want to measure the shape of (let's call it f), and you have a bunch of simpler functions (let's call them g1, g2, g3, etc.), then you can combine these simpler functions in different ways to get really close to the shape of f.
Think of it like building with Legos. You have a bunch of different shapes and sizes, but you can put them together in different ways to build whatever you want. In the same way, you can use the simpler functions to build up the shape of f.
This is really useful because sometimes it's hard to work with complicated functions directly. But with the Stone–Weierstrass theorem, you can break them down into simpler pieces and work with those instead.
So, to sum it up: Stone–Weierstrass theorem says that you can build any function you want using simpler functions, kind of like building with Legos.