Inner product/Cauchy-Schwarz/Angle/Remark

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

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

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

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

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

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

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 has many disadvantages, computationally and in terms of the proofs.