Kakutani's Fixed Point Theorem

Piste

Can someone please explain this theorem in terms I could understand? I Wikied it but the result I got was far too complex for me though I would really like to understand it.

pH

Very simply what it says is that for *certain* functions which map points to points in the same space, there exists a point which maps to itself.

An example of this is as follows:

If you have 2 sheets of paper, both the same size.

- Lay one on top of the other. *All* points on the top sheet are now exactly above the corresponding point on the lower sheet.

Now take the top sheet and crumple it anyway you like, and then place it on the bottom sheet so that it doesn't overlap the edge.

The fixed point theorem says that no matter how you crumple(not tear!) the top sheet, when you put it down there is *at least* one point on the top sheet that is still directly above its corresponding point in the lower sheet.

David19

As pH said, a fixed-point theorem is a result saying that a function F will have at least one fixed point, (a point x for which F(x) = x).

An example would be f(x) = xsquared − 3x + 4 because f(2) = 2.

This is taken from the wikipedia article on fixed point theorem so you could check that out if you haven't.

The Kakutani fixed point theorem is a type of fixed point theorem.

Piste

They're great explainations, thank you so much!

Now I need to know how this corresponds to Infinite Regress but I don't know if that's more Mathematics or Game Theory/Economics...

Son Goku

Just to add to pH and David19, fixed point theorems are usually proved using topology.
It is probably the best example of a problem which requires very little maths to understand, but requires very advanced math to tackle.

(It is often tackled using things called cohomology groups.)