I'll assume the reader has a basic knowledge of the theory of linear spaces. Elementary concepts such as vector, basis, and span will be used without definition, unless a new meaning or interpretation is being invoked.
A Hilbert Space is a complete inner product space.
In other words, a Hilbert Space is a Vector Space, with an inner product, which is complete under the distance function induced by that inner product. That is: the inner product (x,y) induces a norm, ||x||=(x,x) which in turn induces a distance function, d(x,y)=||x-y||. Any sequence which is Cauchy under this distance function must converge to an element of our space.
This is essentially the usual vector space construction used in most descriptive mathematical
endeavor. It provides a system of coordinates to describe the condition or ``State'' of the
system. Accordingly, it will be termed the ``State Space'', the set of all possible values
of the variables
comprising our model. In general, it will be the Euclidean Space 
, with the standard
inner product (ie, the dot product...).
In our first Example, the state space is two-dimensional; the position (the system's state) is restricted to the thin plane of oil lying at the interface of sea and sky. Later, the state space is extended to three dimensions, but the observations remain two dimensional.