Uniqueness (or not) of an inner product on some vector space I have looked through several of the posts here on SE concerning uniqueness and inner products, but they don't seem to be answering my question (granted, I am not a mathematician and have only started looking at a more formal approach to vectors, so they may very well be answering the question but I am missing it!)
From reading some lecture notes, the conditions for an inner product for a real vector space were given (symmetry, linearity in the second term (and for a real ector space also in first term by symmetry), non-negativity and non-degeneracy).
Now in the notes, it says after the non-negativity condition:

P(iii) Non-negativity, i.e. a scalar product of a vector with itself should be positive, i.e.
  $\textbf{a · a} \geq 0 $. This allows us to define
  $|\textbf{a}|^2 = \textbf{a · a}$ ,
  where the real positive number $\textbf{|a|}$ is a norm (cf. length) of the vector a. It follows from the above
  with $\textbf{a} = \textbf{0}$ that
  $\textbf{|0|}^2 = \textbf{0 · 0} = 0$ .

The part$|\textbf{a}|^2 = \textbf{a · a}$ makes it seem to me as if there is only one function that gives the inner product (I can't think how there could more than one function that gives $\textbf{a . a}=|\textbf{a}|^2$, but this may be me being dumb.
I would greatly appreciate if someone could clarify the following:


*

*There can be more than one function that can act as an inner product on a vector space I think? It would be great if someone could give an example of a vector space explicitly and a could of different inner products on it. What use is there for having these different inner products?

*Is it the case that the inner product for the real vector space is unique, hence the confusion from my notes. Or perhaps the statement in my notes doesn't even imply that the inner product for real evctor space is unique?

 A: I think you think that $|a|$ is some known attribute of a vector in a vector space. It's not. The quotation in your question shows that it is defined in terms of the inner product. So different inner products give different lengths.
For example, consider the inner product on $\mathbb{R}^2$ given by
$$
\langle  (x,y),(u,v) \rangle = xu + 2yv .
$$
A: Inner product isn't unique in general : in fact, let's take the real vector space with any inner product $(\mathbb{R}^n, (\cdot | \cdot ))$.
For every self-adjoint oppérator on this space such as $\text{Spec}(u) \in \mathbb{R}^*_+$, we can define another inner product : $(x, y) \mapsto (x | u(y))$. This gives us a lot of new inner products !
More generally, for any real space of basis $\mathcal{B}$, we can define an inner product in which $\mathcal{B}$ is an orthonormal basis : $(x, y) \mapsto \sum_{e \in \mathcal{B}}e^*(x)e^*(y)$ (here, e* is the linear form giving the coordinate associated to $e$, e.g. in $\mathbb{C}$, $\Re = 1^*$ and $\Im = \text{i}^*$).
Hope it helps !
