Dimension of the State Space of an Ideal Gas

There is a subtle difference between space and dimension. A space can have any any number of dimensions, but a subset of that space can have any dimension less than the space itself.
In any case the dimension of a set is the number of numbers needed to define any element of the set.
Consider the set of points  
\[\{ (x,x^2) : x \in \mathbb{R} \}\]
. This set defines the line  
\[y=x^2\]
  which is a curve in the plane. The plane is two dimensional but the curve is one dimensional since only one point  
\[x\]
  is needed to define each point on the curve.
This example shows how it is possible for a space to be embedded in a higher dimensional subspace.
The example is illustrative.  
\[x\]
  and  
\[y\]
  are coordinates on different axes. The coordinates are not independent if one coordinate is a function of the other, or if some coordinates are functions are some other coordinates. Some physical systems are defined by physics properties.
One such is a confined ideal gas. A gas is defined totally by its internal heat energy  
\[U\]
, its temperature  
\[T\]
, its pressure  
\[p\]
  and volume  
\[V\]
.
These quantities are not independent however.
\[U\]
  is directly related to  
\[T\]
  by the equation  
\[U= \frac{3}{2} kT\]
  and we can define  
\[T\]
  and hence  
\[U\]
  in terms of  
\[p\]
  and  
\[V\]
  by the equation  
\[pV=nRT\]

where  
\[n\]
  is the number of mols and  
\[R=8.314 J/mol/K\]
  is the Universal molar gas constant.
\[U\]
  and  
\[T\]
  are not needed to describe the state of a gas. We only need  
\[p\]
  and  
\[V\]
  - or in fact any two of  
\[U, \: T, |; p, V\]
. The dimension of the state space of an ideal gas is 2 and we can plot any state of an ideal gas in the plane.