Let X be a Haudorff Topological Vector Space.
Define the following relation on the collection of all Cauchy filters on X:
$F ∼_R G$ ⇔ $∀U$ nbhood of the origin in $X, ∃A ∈ F, ∃B ∈ G$ s.t. $A − B ⊂ U$.
We define $\hat{X}$ as the quotient of the set of all c.f. on X w.r.t. the equivalence relation (R).
Hence, an element $\hat{x}$ of $\hat{X}$ is an equivalence class of c.f. on $X$ w.r.t. (R).
Let $U$ be an arbitrary nbhood of the origin in X. Define $ \hat{U}:= \{ \hat{x} ∈ \hat{X} : U ∈ F \text{ for some } F ∈ \hat{x}\}$. and consider the collection $ \hat{B} := \{ \hat{U}: U \text{ nbhood of the origin in } X \}$.
The filter generated by $ \hat{B} $ fulfills all the properties in this theorem
A filter $F$ of a vector space $X$ over $K$ is the filter of neighbourhoods of the origin w.r.t. some topology compatible with the vector structure of X if and only if
The origin belongs to every set $U ∈ F$
$∀U ∈ F, ∃V ∈ F$ s.t. $V + V ⊂ U$
$∀U ∈ F, ∀λ ∈ K$ with $λ \ne 0$ we have $λU ∈ F$
$∀U ∈ F$,$U$ is absorbing.
$∀U ∈ F, ∃V ∈ F$ s.t. $V ⊂ U$ is balanced.
How can I check this?
I try to do this but cant because I am new reader in Topological Vector spaces.
Please help me.