How to motivate the axioms for the inner product Typically, one doesn't just write down lists of axioms and then sees
if there are enough interesting examples that satisfy them; they evolve
over time, usually from a couple of very important/interesting examples
that are then generalized.
For the vector space axioms, for example, it is pretty easy to motivate
them because they crop up everywhere and are easily spotted, so it
is "natural" (as natural as mathematics can be...) to write
them down in an abstract fashion and say "now we are just going
study just what follows from these axioms".
But for the axioms of a certain class of maps from a pair of vector
spaces to (to make it simple) $\mathbb{R}$, namely the inner product,
I don't find their motivation satisfying at all. From what I read
in some books and Wikipedia everything boils down to saying: 1) It
is a geometric fact that in - for simplicity - $\mathbb{R}^{2}$ the
equation 
$$
\left\langle x,y\right\rangle =\left\Vert x\right\Vert \left\Vert y\right\Vert \cos\theta\quad\quad\left(1\right),
$$
holdS, where $\theta$ is the angle between $x,y$ and $\left\langle \cdot,\cdot\right\rangle $
is defined as the dot product.
2) $\left\langle \cdot,\cdot\right\rangle $ has the properties of
being symmetric, linear in each argument and positiv definite.
3) Conclusion: We should abstractly study symmetric, linear and positiv
definite maps $V\times V\rightarrow\mathbb{R}$, where $V$ is a vector
space.
For me, 1) and 2) aren't by far enough to say 3), since 
$\bullet$ for other important examples of maps (and vector spaces
$V$), the relation $\left(1\right)$, which motivated the abstract
definition of an inner product, isn't applicable at all: It isn't
intuitively clear what $\left\langle \cdot,\cdot\right\rangle $ and
$\theta$ should be for these examples, so that we can verify $\left(1\right)$
for them, observe that in all these examples the LHS has the properties
listed in 2), which would consolidate our belief that we truly have
carved out an important class of mappings that is worthwhile studying
in the abstract. Consider e.g. $V=C\left[a,b\right]$ and 
$$
\left(x,y\right)\mapsto\int_{a}^{b}x\left(t\right)y\left(t\right)dt.
$$
$\bullet$ there are a ton of other properties the dot product $\left\langle \cdot,\cdot\right\rangle $
has. Why not study maps that satisfy some other geometric intuitive
properties besides the ones in 2) ?
So what I think I'm searching for is a better motivation of the axioms
of the inner product or for more example (that are qualitatively different
from another) satisfying $\left(1\right)$. 
Note bene: Trying to motivate the axioms of the inner product by its
history didn't bring me much clarity: All I could find after some
googling was that the definition of the dot product came from the
definitions of the quaternions (see History of dot product and cosine),
but going from there to defining inner products abstractly seems to
be a bit stretched for me.
 A: Symmetry, bilinearity, and positive definiteness are exactly the properties used to prove the Cauchy-Schwarz inequality.  Well, there are a zillion proofs of the Cauchy-Schwarz inequality; I mean the one that proceeds by observing $0\le \|x-ty\|^2$ for all $t\in\mathbb R$, expanding to obtain a quadratic in $t$, and concluding that the discriminant of that quadratic is nonpositive (and then you fiddle with definiteness to get the equality case).
In other words, an inner product is just a map for which that proof is correct.
We want to obtain the Cauchy-Schwarz inequality in other spaces because it's a cornerstone of the linear-algebraic treatment of Euclidean geometry — you use it to prove the triangle inequality, to show that orthogonal projections are metric projections (which gets you everything you want to know about tangent planes to spheres), etc.  (The equation (1) is part of all that: in this treatment, it's essentially the definition of angle.  You need Cauchy-Schwarz to show that it's well-defined.)
A: (Based on Qiaochu Yuan's comment above) It is natural to study normed spaces as by imparting distance we get additional structure in a vector space which allows us to explore its geometric properties. However the class of normed spaces is by itself also somewhat large. We may also be motivated to add additional structure to have orthogonality in normed spaces (so that a lot of nice things can happen: for example one can find the coordinates of a vector more efficiently). 
To do so one takes a generalization of the Pythagoras theorem (parallelogram law) and isolates those normed spaces which satisfy it. The polarization identity now imparts the necessary orthogonality structure. By the Fréchet–von Neumann–Jordan theorem these are precisely the spaces isolated by the inner product axioms.
