dot product and relationship to cosine of its angle In our class, we defined that the term  $$ \cos(\alpha)=\frac{\langle x,y \rangle}{\sqrt{\langle x,x\rangle \langle y,y \rangle}}$$ is equal to the cosine of the angle enclosed by $x$ and $y$. However, we restricted ourself to Euclidean vector spaces.
Now I am wondering: is this definition is reasonable for all unitary vector spaces too?
 A: In a unitary vector space, you can certainly define this, but it is (1) not necessarily real, and therefore not necessarily the cosine of an angle, and (2) it isn't symmetric, since $\left<x,y\right>\neq \left<y,x\right>$ in general.  
If you do define $$\cos(x,y)=\frac{\left<x,y\right>}{\sqrt{\left<x,x\right>\left<y,y\right>
}}$$ you get $|\cos(x,y)|\leq 1$ and $\cos(y,x)=\overline{\cos(x,y)}$, and $\cos(\alpha x, x)=\frac{\alpha}{|\alpha|}$ for $\alpha\in\mathbb C\setminus\{0\}$.
So you can use this as a generalized notion of the cosine function.
See: http://en.wikipedia.org/wiki/Inner_product_space for some other ideas. There, they define the "angle" between complex vectors in $[0,\pi/2]$ as:
$$\text{angle}(x,y)=\arccos |\cos(x,y)|$$
The real part of $\cos(x,y)$ plays the part of of $\cos$ in a unitary "law of cosines."
$$\|x+y\|^2 = \|x\|^2+\|y\|^2 + 2\|x\|\|y\|\Re \cos(x,y)$$
Indeed, if you take $\mathbb C^n$ and give it the "standard" complex inner product, the real part of this inner product is just that "standard" inner product on $\mathbb R^{2n}$. 
The imaginary part of this dot product is the sum of the (oriented) areas of the parallelograms formed by the corresponding pairs of complex numbers.  That gives some intuition, perhaps, for the geometry of the "complex part" of the unitary dot product.
