What is it about some (or all) of the properties of Hilbert Spaces that makes them so powerful/necessary when modeling so many different systems? Tried this question on the Physics side, but apparently it was off topic, and it seemed as though it was rather badly asked, so let me try to do better here:

To be a Hilbert Space, your space $(V,+,\cdot)$ needs to:
Be a Vector Space:
$ \forall u,v,w\in V $ and $a,b\in\mathbb{F}$
1) Associativity of addition -- $u+(v+w)=(u+v)+w$
2) Commutativity of addition --- $u+v=v+w$
3) Identity Element of Addition -- $\exists 0\in V:\forall v\in V, v+0=0$
4) Inverse Element of Addition -- $\forall v\in V \exists -v\in V: v+(-v)=0$
5) Associativity with scalar multiplication -- $a(bv)=(ab)v$
6) Identity Element of Scalar Multiplication -- $1v=v$ for $1\in F$
7) Distributivity of scalar multiplication w.r.t vector addition -- $a(u+v)=au+av$
8) Distributivity of scalar multiplication w.r.t scalar addition -- $(a+b)v=av+bv$
Have an Inner Product
$\exists \langle\cdot,\cdot\rangle:V \times V \to \mathbb{F}$ with the properties:
1) Conjugate Symmetry -- $\langle v,w\rangle = \overline{\langle w, v \rangle}$
2) Linearity in the First Argument: $ \langle au, v\rangle=a\langle u,v\rangle$ and $\langle u+v, w\rangle = \langle u,w \rangle + \langle v, w \rangle$ 
3) Positive Definiteness -- $\langle v, v \rangle \geq 0$ and $\langle v,v\rangle = 0 \iff v=0$
Be Complete
If a series of vectors converges absolutely, i.e.:
$ \sum_{n=0}^{\infty} ||v_n||<\infty$
Then the series converges in the space V. 

These spaces, and the elements of them, can have vastly different forms -- n-d vectors and with the normal inner product, functions with the integral inner product, etc. We can used spaces like this to model a vast array of physical systems using Vector Analysis, Sturm-Liousville Theory, Fourier Analysis, Spherical Harmonics, etc. These systems solve problems ranging from Classical to Statistical to Quantum Mechanics.
Now, by no means would I expect anyone to list why all of these properties are important, but:

What is it about some of the particular properties of Hilbert Spaces that makes them so useful? 

Even specific examples of how these particular properties (as opposed to not having them) make Hilbert Spaces powerful in particular cases would be nice.
For instance, why is Conjugate Symmetry of the inner product of two elements of the space a necessary part of the structure of the space when it comes describing so many different systems? Or Linearity in the First Argument? For that matter, why should all Cauchy sequences of elements be required to converge in the space for it to be a useful modeling structure.
What is it about a complete, (usually) complex inner-product space that makes it so useful in so many ways?
Sorry if this question doesn't make much sense, I'll try to do better in the comments if there are any suggestions as to how to make it more clear what I'm asking.
 A: One critically important property of Hilbert space $H$ is this: If $V$ is a closed convex subset of a Hilbert space, then, for every $x \in H$, there exists a unique $v \in V$ that is closest to $x$. This property implies the Riesz representation theorem for bounded linear functionals, and this property also leads to well-defined orthogonal projection onto a closed subspace. Such existence and uniqueness allows one to obtain unique solutions to differential equations, and to various other equations by dealing with functions as points in a space with an infinite-dimensional Euclidean geometry.
Hilbert space connects sums of squares (such as energy) with Geometry, and allows one to analyze problems by either using orthogonal projection or closest-point projection, which are the same for closed subspaces. Infinite sets of equations in infinitely many variables are then solved using geometry, closest point, and/or iterative successive approximation. Algebra and Analysis are almost equivalent in this setting. The real power in these ideas becomes evident when one applies such techniques to spaces where the points are complicated objects such as functions. Hilbert space grew out of studying Fourier series and integrals, and other orthogonal function expansions, which makes Hilbert space a natural fit for dealing with functions, and for solving differential equations.
