# Proof of the Mazur-Ulam Theorem

The Mazur-Ulam Theorem Theorem $$2.1$$ states that any surjective isometry between any two real normed spaces $$f:X \rightarrow Y$$ is affine.

In the proof of the theorem, the author mentioned that it suffices to show that for any $$x, y \in X$$, $$f\left(\dfrac{x+y}{2}\right) = \dfrac{f(x)+f(y)}{2}$$

Why is it the case? How to conclude $$f$$ is affine from equation above?

Recall that $$f:X \rightarrow Y$$ is an affine function if for all $$x,y \in X$$ and $$0 \leq t \leq 1$$, $$f[(1-t)x+ty] = (1-t)f(x) + t f(y)$$

The midpoint-affine property $$f\left(\dfrac{x+y}{2}\right) = \dfrac{f(x)+f(y)}{2} \tag{1}$$ implies being affine under the assumption that $f$ is continuous (which it is, being an isometry). As stated, $(1)$ amounts to the case $t=1/2$ of $$f[(1-t)x+ty] = (1-t)f(x) + t f(y)\tag{2}$$ But applying $(1)$ again, the second time to $x$ and $(x+y)/2$, yields $(2)$ for $t=1/4$. Similarly, applying $(1)$ to $y$ and $(x+y)/2$ yields $(2)$ for $t=3/4$.
Continuing this process, we obtain $(2)$ for all dyadic rationals in $(0,1)$: numbers of the form $k/2^m$, $0<k<2^m$. These are dense in $[0,1]$ and since both sides of $(2)$ are continuous with respect to $t$, equality $(2)$ holds for all $t\in [0,1]$.
If it is given that $$f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$$ Then $$f\left(\frac{x+y+z+w}{4}\right)=\frac{f(\frac{x+y}{2})+f(\frac{z+w}{2})}{2} =\frac{f(x)+f(y)+f(z)+f(w)}{4}$$ So by induction, $$f\left(\frac{1}{2^n}\sum_{i=1}^{2^n}{a_i}\right) = \frac{1}{2^n}\sum_{i=1}^{2^n}{f(a_i)}$$ And any real number between 0 and 1 can be estimated by something over $2^n$ for large enough $n$.