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

by AxiomOfChoice
Tags: base, exact, local, meaning, space, topological, vector
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AxiomOfChoice
#1
Mar18-11, 02:04 PM
P: 529
I am confused as to exactly what a local base at zero (l.b.z.) tells us about a topology. The definition given in Rudin is the following: "An l.b.z. is a collection G of open sets containing zero such that if O is any open set containing zero, there is an element of G contained in O". Ok, great.

But I have seen some proofs in my functional analysis class that suggest something like the following: Any open set in the topology can be formed by taking unions (possibly uncountable) of *translations* of sets in a l.b.z. Is this true, or am I just missing something?
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micromass
#2
Mar18-11, 02:08 PM
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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 [tex]V\subseteq G-a[/tex]. Thus [tex]a+V[/tex] contains a and is smaller than G. Now, we can write G as

[tex]G=\bigcup_{a\in G}{a+V_a}[/tex]

Thus we have written G as union of translations of the lbz...
AxiomOfChoice
#3
Mar18-11, 02:11 PM
P: 529
Quote Quote by micromass View Post
Yes, the two are equivalent!
Great, thanks! Now that I know that, I'm going to try to work out a proof. But is this discussed in Rudin, or on the web, somewhere in case I get stuck?

micromass
#4
Mar18-11, 02:14 PM
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Exact meaning of a local base at zero in a topological vector space...

Sorry I posted too fast. I was going to include a proof. I've edited my post 1 with the proof...
Landau
#5
Mar18-11, 03:44 PM
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"
[itex]T_g:x\mapsto x+g[/itex]
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.


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