Let \(V\) be a vector space over R or C with an inner product \({\langle \;, \;\rangle} \). For any vectors \(v\) and \({w \in V}\)
$$ \lVert v \rVert \lVert w \rVert \geq \left| \langle v , w \rangle \right|$$

Proof:
For \(v= 0\) it's obvious that \(\lVert v \rVert \lVert w \rVert = 0 =\left| \langle v , w \rangle \right|\)

For \(v\neq 0\) the projection of the vector \(w\) onto the plane orthogonal to \(v\): \(w_{ \perp }=w-v\frac{ \langle v,w \rangle }{ \langle v , v \rangle } \)
$$\langle v,w_{ \perp } \rangle = \langle v ,w \rangle - \langle v , v \rangle \frac{ \langle v,w \rangle }{ \langle v , v \rangle } =0 $$
$$\lVert w \rVert^2=\lVert w_{ \perp } \rVert^2+ \frac{\left| \langle v,w \rangle \right|^2 }{ \lVert v \rVert^2 } \geq \frac{\left| \langle v,w \rangle \right|^2 }{ \lVert v \rVert^2 } $$
$$ \lVert v \rVert \lVert w \rVert \geq \left| \langle v , w \rangle \right|$$