# Exact meaning of a local base at zero in a topological vector space...

by AxiomOfChoice
Tags: base, exact, local, meaning, space, topological, vector
 Mentor P: 18,062 Yes, the two are equivalent! Basically, take an open set G in the topology. If a is in G, then G-a contains 0, thus we can find an element V_A of the lbz, such that $$V\subseteq G-a$$. Thus $$a+V$$ contains a and is smaller than G. Now, we can write G as $$G=\bigcup_{a\in G}{a+V_a}$$ Thus we have written G as union of translations of the lbz...
 Sci Advisor P: 905 You are already familiar with a neighbourhood base in any topogical space. Now, the topology on a t.v.s. (or a topological group for that matter) is translation-invariant. This is because "translation by a fixed g" $T_g:x\mapsto x+g$ is a homeomorphism (which is because addition is by definition continuous, and T_g is obviously invertible). So it suffices to consider the neighborhood base of any point, in particular 0.