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.

We can used spaces like this to model a vast array of physical systems. Sturm-Liousville Theory and Fourier Analysis rely on the properties of Hilbert Spaces of functions with an inner product defined using integration. 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?

For instance, why is Conjugate Symmetry of the inner product of two elements of the space necessary for describing so many systems? Or Linearity in the First Argument? For that matter, why should all Cauchy sequences of elements be required to converge in the space?

What is it about a complete, (usually) complex inner-product space that makes it so useful in so many ways?

P.S. (but not all that important): What exactly is meant when it says that a Hilbert Space generalizes the notion of Euclidean Space?

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    $\begingroup$ You last part: Hilbert space expands calculus and vector algebra into any number of dimension, ( even, or especially perhaps, infinite dimensions) based orginally on the simplest, most straightforward 2 dimensional Euclidean space ideas and axioms. Personally, I think this is more a mathSE type question, as you cover a lot of ground, all based on math concepts. Best of luck with the question anyway. $\endgroup$ – user108787 Aug 12 '16 at 21:02
  • $\begingroup$ The Hilbert space is simply the infinite dimensional harmonic oscillator. Why is it so useful? For the same reason that the harmonic oscillator is useful: it can be calculated with relatively little effort. $\endgroup$ – CuriousOne Aug 13 '16 at 1:09
  • $\begingroup$ We all agree that Hilbert Spaces as useful and easy to compute with, but my question is more about the specific properties of Hilbert Spaces (like conjugate symmetry with respect to the inner product, or completeness) that make them so useful, and specifically why those properties are useful in modeling so many different systems. What is it about the particular set of properties that Hilbert Spaces have that make them such natural mathematical spaces in which to model $\endgroup$ – D. W. Aug 13 '16 at 1:14
  • $\begingroup$ Why does nature make a lot of harmonic oscillators? It doesn't. We are the ones who are fitting them to everything and anything that we see and we hope that they will work well enough to be useful (whenever they are not, the pain to calculate things goes up a hundred fold). I am getting the feeling that your model of what physics is is not a very precise rendition of what physics actually is. $\endgroup$ – CuriousOne Aug 13 '16 at 1:59
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    $\begingroup$ I feel you can ask the same question about perturbation theory, Fourier transforms and Taylor expansions, (and conformal transformations using complex numbers). We take the math we have and adapt it. If we really live in a mathematical based reality, then it makes sense to me. $\endgroup$ – user108787 Aug 13 '16 at 10:16