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?

Usually this is how one hears the definition anyway: "a subspace of codimension 1." 
So yes, I believe that any subspace $W$ of a vector space $V$ can be called a hyperplane if $\dim(V/W)=1$. 

(Added) 
This is, of course, equivalent to saying that $W$ is the kernel of a linear functional (if you want to do it that way.) 
If $\dim(V/W)=1$, then picking a basis for $W$ and extending it by one element "$v$" to be a basis for $V$ allows you to project onto the coefficient of $v$, obtaining a linear functional from $V$ to $F$ with kernel $W$.
In the other direction, an easy application of isomorphism theorems tells you that for a nonzero linear functional $f$, $V/\ker(f)\cong F$, so $V/\ker(f)$ has dimension $1$, and $\ker(f)$ has codimension $1$.
