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?

I know this is an old question, but it seems to me that no one has answered it in a "correct" fashion yet, concerning "infinite" of course.
The most generalized definition I've seen is the next one:
Let H be a subspace in a vector space X. H is called hyperplane if $H \neq X$  and for every subspace V such that $ H \subseteq V $ only one of the following is satisfied: $ V = X$ or $ V = H $.
Lemma: A subspace H in X is hyperplane iff e is in X \ H such that $$ <\{e,H \}> =  \{ \lambda e + h : \lambda \epsilon \mathbb{R}, h  \epsilon H\} = X  $$
Just for fun:
Theorem: A subspace H in a vector space X is hyperplane iff there is a non-zero linear function $$ l : X \to \mathbb{R}$$ such that $$ H = \{x \epsilon X : l(x) = 0\} = Ker( l ) $$
Good luck proving this.
