Hyperplanes in finite and infinite dimension vector spaces. I know that hyperplanes of a n-dimensional vector space are sub-spaces of dimension n-1, This is in finite dimension spaces. BUT what about infinite dimension spaces what are hyperplanes? are they the same?

When you are generalizing an idea to a domain in which it is not, a priori defined, you can do it in any way you want. 
If your definition of hyperplane is that it is a subspace of dimension $n-1$ where $n$ is the dimension of the space, and now you want to extend this for when $n$ is $\infty$, you could say that $\infty-1=\infty$ and therefore you will call hyperplane any subspace of infinite dimension. 
The thing is that often one generalizes when there is a need for it. When you can find something interesting in the generalization. 
For finite dimension a hyperplane is the zero set of a linear form (a linear functional), a linear function from the space to the scalar field. So, another possibility is to say that a hyperplane is the zero set of a linear functional. This is the option that is more often used.
A rough criterion of how good a definition is Halmos': A good definition is the hypothesis of a theorem. The thing is that many theorems involve the zero set of linear functionals. 'any infinite dimensional subspace' are less interesting as a class. 
