Re: Brouwer's Fixed Point Theorem

On Feb 4, 8:41 pm, "avital" <avitaloli...@xxxxxxxxx> wrote:
Hi,

What questions or ideas lead to the relevance of Brouwer's Fixed Point
Theorem? What I mean is, what do we understand better once it's
proven? I'd prefer to know of the things which caused Brouwer to
originally think about the question and prove it rather than hyper-
modern examples of applicability.

Don't know about Brower's original motivation. I do know that fixed
point theorems in general tell us about the existence of solutions of
equations.

For instance, Brower's theorem says that given a continuous map f: D -
D, where D is the n-dimensional unit ball, then the equation f(x) =
x has a solution.

In spirit, this theorem is similar to the intermediate value theorem.
One way to state it is that for any continuous map f: I -> I, where I
is the unit interval, which is positive at 0 and negative at 1, there
exists a solution of the equation f(x) = 0.

Hope this helps.

Igor

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