Inner product/Cauchy-Schwarz/Angle/Remark

For vectors ${}v$ and ${}w$, different from ${}0$, in a euklidean vector space ${}V$, the inequality of Cauchy-Schwarz implies that

${}-1\leq {\frac {\left\langle v,w\right\rangle }{\Vert {v}\Vert \cdot \Vert {w}\Vert }}\leq 1\,$

holds. Using the trigonometric function cosine (as a bijective mapping ${}[0,\pi ]\rightarrow [-1,1]$) and its inverse function, the angle between the two vectors can be defined, by setting

${}\angle (v,w):=\arccos {\frac {\left\langle v,w\right\rangle }{\Vert {v}\Vert \cdot \Vert {w}\Vert }}\,.$

The angle is a real number between ${}0$ and ${}\pi$. The equation above can be read as

${}\left\langle v,w\right\rangle =\Vert {v}\Vert \cdot \Vert {w}\Vert \cdot \cos {\left(\angle (v,w)\right)}\,.$

This provides the possibility to define the inner product in this way. However, then we have to find an independent definition for the angle. This approach might look a bit more intuitive but is has computationally and in terms of the proofs many disadvantages.