Let $(E, \vert \vert \cdot \vert \vert )$ be a normed space. The norm is induced by a scalar product iff in $(E, \vert \vert \cdot \vert \vert )$ the Apollonios equation holds.
The Apollonios equation: $\frac{1}{2} \vert \vert x-y \vert \vert ^2 + 2\cdot \vert \vert z -\frac{1}{2}(x+y)\vert \vert ^2 = \vert \vert z-x \vert \vert ^2+\vert \vert z-y \vert \vert ^2$
So I have to prove: $\vert \vert x\vert \vert =\sqrt{\langle x,x\rangle} \Longleftrightarrow \frac{1}{2} \vert \vert x-y \vert \vert ^2 + 2\cdot \vert \vert z -\frac{1}{2}(x+y)\vert \vert ^2 = \vert \vert z-x \vert \vert ^2+\vert \vert z-y \vert \vert ^2$.
The direction $\Rightarrow$ was easy. How do I prove $\Leftarrow$? My idea was to show that the properties of the norm and the scalar product are the same. But it seems to be too complicated. Is there any easy way to prove it?
If someone gives you a normed space, it is quite a challenge to come up with a definition of the inner product. Sure, $\langle x, x \rangle$ is easy for every $x$, but what about $\langle x, y \rangle$? If you can come up with the right definition of $\langle x, y \rangle$ in terms of the norm, then you are half way there. Actually this is the interesting, creative half.
But then you still need to verify that the function $\langle , \rangle$ from $E \times E$ to $\mathbb{R}$ has indeed the three or so properties that make it an inner product. This is the boring, routine verification half, but that is where you need Apollonios.