Brouwer Fixed Point Theorem

Brouwer Fixed Point Theorem

The Brouwer Fixed Point Theorem states that any continuous function from a convex, compact set in Euclidean space to itself has a fixed point, i.e., a point that is mapped to itself by the function.

To prove this theorem, we will use a proof by contradiction. Suppose that there exists a continuous function f from a convex, compact set K in Euclidean space to itself that does not have a fixed point. Then, for any point x in K, we have f(x) is not equal to x.

Now, consider the function g(x) = ||f(x) - x||, where ||.|| denotes the Euclidean norm. Since f is continuous and K is compact, g is also continuous and attains its minimum value at some point y in K. That is, g(y) = ||f(y) - y|| is the smallest possible value of g over all points in K.

Since f(y) is not equal to y, we know that g(y) is greater than zero. However, we can also show that g(f(y)) is strictly less than g(y). To see this, note that g(f(y)) = ||f(f(y)) - f(y)|| = ||f(y) - y|| < ||f(y) - y|| = g(y), where the strict inequality follows from the fact that f(y) is not equal to y.

This contradicts the fact that g(y) is the minimum value of g over all points in K, since g(f(y)) is strictly less than g(y). Therefore, our initial assumption that f does not have a fixed point must be false, and we conclude that any continuous function from a convex, compact set in Euclidean space to itself has a fixed point.

In summary, the Brouwer Fixed Point Theorem states that any continuous function from a convex, compact set in Euclidean space to itself has a fixed point, and we have proven this theorem using a proof by contradiction.