Mazur-Ulam theorem

If we have map between two Banach spaces which preserves distances, then it has to be a linear! (Affine)

Let a: X \to Y be the map. To prove that a is affine it is sufficient to show that a(\frac{x+y}{2})= \frac{a(x)+a(y}{2}.

Consider the defect
\displaystyle \Delta(a) \left \| {a(\frac{x+y}{2})-\frac{a(x)+a(y)}{2}}\right \|
which is bounded by
\displaystyle \left \|a(\frac{x+y}{2})-\frac{a(x)}{2}\right \|+ \left \|a(\frac{x+y}{2})-\frac{a(y)}{2}\right \| \le \frac{\left\| x-y\right\|}{2}

Construct another isometry \displaystyle a' = a^{-1}ra : X \to X where r(z) = a(x)+a(y)-z is a reflection in the Y space about the point \displaystyle \frac{a(x)+a(y)}{2}.

\displaystyle \Delta(a')= \left \| {a'(\frac{x+y}{2})-\frac{a'(x)+a'(y}{2}}\right \| = \left\| a^{-1}(a(x)+a(y)-a(\frac{x+y}{2}))-\frac{x+y}{2} \right\|
\displaystyle = \left\|a(x)+a(y)-a(\frac{x+y}{2})-a(\frac{x+y}{2})\right\| = 2\Delta(a')

Thus if the defect of a is non-zero we can iterate to get arbitrarily large defects. But we saw that the defect has to be bounded by \frac{\left\| x-y\right\|}{2}. Done!

This result falls in the general theme of trying to understand geometry of Banach spaces- how much of the linear structure is captured by purely metric notions? What are different Banach space concepts? How are different Banach spaces related- Do they locally look similar? Can we embed a metric space into another with some deformation?

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