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A locally convex space is called Barrelled if each closed absorbing convex set is 0-neighborhood See. But i doubt that every absorbing set contains zero. Then is every LCV is barreled. I think, i am confusing with something...

Every absorbing set contains zero. Because if $A$ is absorbing set then there is some $t>0$ such that $0\in tA$, that is we have $0= t.a$ for some $a\in A$. This gives $a=0\in A$.

So My question is:

If every absorbing set contains zero, then it means that every LCV is barreled.. so what is need to defined barreled in LCV.. can i get some topological vector space which is not barreled

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  • $\begingroup$ Sorry, but I do not get what the actual question is. $\endgroup$ – Michael Greinecker Jan 7 '13 at 11:09
  • $\begingroup$ Sorry.. I am editing the question.. My question is if every absorbing set contains zero, then it means that every LCV is barreled.. so what is need to defined barreled in LCV.. can i get some topological vector space which is not barreled.. $\endgroup$ – zapkm Jan 7 '13 at 11:12
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    $\begingroup$ In order for $A$ to be a $0$-neighborhood you need more than $0 \in A$. You need an entire open set $U \subset A$ such that $0 \in U$. $\endgroup$ – Martin Jan 7 '13 at 11:16
  • $\begingroup$ @Martin. Thanks i got it... condition is CLOSED absorbing set contains zero Neighborhood.. $\endgroup$ – zapkm Jan 7 '13 at 11:19
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Take the subspace $X$ of $C[0,1]$ consisting of continuous functions $f \colon [0,1] \to \mathbb C$ vanishing in a neighborhood of $0$. That is, for every $f \in X$ there exists $\delta \gt 0$ such that $f(t) = 0$ for all $t \in [0,\delta)$.

Equip $X$ with the supremum norm, so $X$ is locally convex. [Note that $X$ is not closed in $C[0,1]$: it is dense in the subspace of functions vanishing at $0$, but $\sqrt{x} \notin X$. It is necessary to have a non-complete space for an example involving a normed space, since Banach spaces (and more generally, Baire locally convex spaces) are barreled.]

The set $$B = \{g \in X : |g(1/n)| \leq 1/n \text{ for all } n \in \mathbb{N}\} \subset X$$ is a barrel: By definition it is closed, convex and circled and $0 \in B$. Moreover, $B$ is absorbent since every function $f \in X$ vanishes in a neighborhood of zero, so multiplying $f$ by a suitably small constant $\lambda$ ensures $\lambda f \in B$.

On the other hand, $B$ is not a neighborhood of zero:

For every $\varepsilon \gt 0$ there is $N \in \mathbb{N}$ such that $1/N \lt \varepsilon$; take $f_N\colon [0,1] \to [0,1/N]$ to be piecewise linear such that $f = 0$ on $[0,1/3N]$ and $f= 1/N$ on $[1/2N,1]$. This a function $f_N \in X$ which is in the $\varepsilon$-neighborhood of $0$, but $f(1/2N) = 1/N > 1/2N$, so $f_N \notin B$.

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  • $\begingroup$ @ martin,I find these concepts of tvs very slippery just like this.can you suggest where I can find a clear exposition of this. $\endgroup$ – Koushik Jan 7 '13 at 12:22
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    $\begingroup$ @K.Ghosh I like Trèves's book on topological vector spaces. Other good references include Rudin's functional analysis, or the books on topological vector spaces by Grothendieck, Kelley-Namioka, Köthe or Schaefer. $\endgroup$ – Martin Jan 7 '13 at 12:34
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6 years later (or too late) just a hint: http://en.wikipedia.org/wiki/Barrelled_space has another example of a normed vector space which is not barrelled: $L^2([0, 1])$ topologized as a subspace of $L^1([0, 1])$. Here the $L^2$ unit ball $B^2$ is a barrel: its integer multiples $nB^2$ cover the entire $L^2$. But $B^2$ is not a neighborhood of $0$ in the $L^1$ norm topology, it does not contain an $L^1$-open set.

As a normed vector space not barrelled, this is another non-complete example, which is necessary as @Martin pointed out.

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