The real part treated like an angle in complex vector spaces In my current lecture I regularly encounter usage of the real part of, say, a scalar product of two vectors similar to angles in classical geometry. For example in Hilbert space theory:
Let $H$ be a Hilbert space, $C \subset H$ be convex and closed. For $x_0 \in H$, $x \in C$ is the best approximation of $x_0$ by $C$, iff
$\forall y \in C : Re \langle x - x_0, y - x_0 \rangle \leq 0$
As much as it is intuitive (and the proof itself is no problem), I do not know how to interpret this. So the question (not necessarily connected to the above example theorem) is
Is there any useful interpretation to the real part of a scalar product in complex vector spaces?
if there is none, in best case the real part is just used for convinience, and the author of my book wants to grasp a more general concept. - in worst case, not such interpretation exists.

I believe it is a notion similar to that of angle in real spaces, but you can't quite call it so. However, I've seen some authors do that.
The relationship between angles and lengths in real vector spaces is given by the formula
$$\lVert v+w \rVert^2 = \lVert v \rVert^2 + \lVert w \rVert^2 + 2\langle v, w\rangle.$$
Indeed, if we define 
$$\cos \theta_{v, w} \lVert v \rVert \lVert w \rVert= \langle v, w \rangle,$$ 
we recover the familiar law of cosines 
$$\lVert v+w \rVert^2=\lVert v \rVert^2 + \lVert w \rVert^2 +2\cos \theta_{v, w} \lVert v\rVert \lVert w \rVert.$$
This last formula holds true in a complex vector space if we define 
$$\cos \theta_{v, w} \lVert v \rVert \lVert w \rVert= \text{Re} \langle v, w \rangle.$$ 
This is one reason why we might call "angle" this number $\theta_{v,w}$. Another reason is given by Arturo's observation above: for example, the formula 
$$ \text{Re} (v \overline{w}), \quad v, w \in \mathbb{C}$$
gives us the real dot product if $\mathbb{C}$ is identified to the Euclidean plane. 
However, there are drawbacks: the most striking (IMHO) is that parallel vectors can have zero angle. Take any non null $v$ and put $w=i v$. Then $\text{Re}\langle v, w \rangle=0$, but would you call them orthogonal?
