Theorem 3.6 i) here (Pratulananda Das and Ekrem Savas: On II-convergence of nets in locally solid Riesz spaces, Filomat 27:1 (2013), 89â€“94, DOI:10.2298/FIL1301089D)

3.63.6 i) Iد„−limsخ±=x0â‡′Iد„−lima.sخ±=a.x0I_د„-\lim s_خ± = x_0 â‡’ I_د„-\lim a.s_خ± = a.x_0 for each a âˆˆ \mathbb Ra âˆˆ \mathbb R

proof: Let UU be a د„د„-neighborhood of zero. Choose V âˆˆ N_{sol}V âˆˆ N_{sol} such that V âٹ‚ UV âٹ‚ U. Since I-\lim s_خ± = x_0I-\lim s_خ± = x_0,

\{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ V\} âˆˆ F(I)\{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ V\} âˆˆ F(I).

Let |a| â‰¤ 1|a| â‰¤ 1. Since VV is balanced, s_خ± âˆ’ x_0 âˆˆ Vs_خ± âˆ’ x_0 âˆˆ V implies that a (s_خ± âˆ’ x_0) âˆˆ Va (s_خ± âˆ’ x_0) âˆˆ V.

Hence we have

\{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ V\} âٹ‚ \{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ V\} âٹ‚ \{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\}\{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ V\} âٹ‚ \{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ V\} âٹ‚ \{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\}

and so

\{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\} âˆˆ F (I).\{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\} âˆˆ F (I).

Now let |a| > 1|a| > 1 and as usual let [|a|][|a|] be the smallest integer greater than or equal to |a||a|. There exists a \color{blue}{W âˆˆ N_{sol} \text{

such that [|a|] W âٹ‚ V}}\color{blue}{W âˆˆ N_{sol} \text{

such that [|a|] W âٹ‚ V}}. Since I_د„-\lim s_خ± = x_0I_د„-\lim s_خ± = x_0,

A = \{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ W\} âˆˆ F (I)A = \{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ W\} âˆˆ F (I) .

Then we have

|a.x_0 âˆ’ a.s_خ±| = |a| |x_0 âˆ’ s_خ±| â‰¤ [|a|] |x_0 âˆ’ s_خ±| âˆˆ [|a|] W âٹ‚ V âٹ‚ U|a.x_0 âˆ’ a.s_خ±| = |a| |x_0 âˆ’ s_خ±| â‰¤ [|a|] |x_0 âˆ’ s_خ±| âˆˆ [|a|] W âٹ‚ V âٹ‚ U

for each خ± âˆˆ Aخ± âˆˆ A. Since the set VV is solid, we have a.s_خ± âˆ’ a.x_0 âˆˆ Va.s_خ± âˆ’ a.x_0 âˆˆ V and so a.s_خ± âˆ’ a.x_0 âˆˆ Ua.s_خ± âˆ’ a.x_0 âˆˆ U for each خ± âˆˆ Aخ± âˆˆ A. So we get

\{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\} âٹƒ A\{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\} âٹƒ A

and so it belongs to F(I)F(I). Hence I_د„-\lim a.s_خ± = a.x_0I_د„-\lim a.s_خ± = a.x_0 for every a âˆˆ \mathbb Ra âˆˆ \mathbb R

Now the \color{blue}{ coloured}\color{blue}{ coloured} part , how do we find that WW? From the definition of N_{sol}N_{sol} we can say we have V_1V_1 s.t V_1+V_1\subset WV_1+V_1\subset W and then V_2\in N_{sol}V_2\in N_{sol} s.t. V_2+V_2\subset V_1V_2+V_2\subset V_1 and thus V_2+V_2+V_2+V_2\subset WV_2+V_2+V_2+V_2\subset W and so on. going this way we can, at one stage, find that nV\subset WnV\subset W where nn is even number. What happens if |[a]||[a]| is odd?

Hope I could make my problem clear to you. pls help.thanks.

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1 Answer

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You know that for each kk there exists VV such that

2^k V \subseteq W.2^k V \subseteq W.

(This can be done in the way you described.)

Now you can simply take kk large enough to have n\le 2^kn\le 2^k and you get

n V \subseteq 2^k V \subseteq W.n V \subseteq 2^k V \subseteq W.

(Just notice that if V\ni 0V\ni 0 and s\le ts\le t are positive integers, then sV\subseteq tVsV\subseteq tV. If we apply this for s=ns=n and t=2^kt=2^k, we get the above inclusion.)1

However, this is used to prove that \newcommand{\Ilim}{\operatorname{I-lim}}\Ilim s_\alpha =x_0\newcommand{\Ilim}{\operatorname{I-lim}}\Ilim s_\alpha =x_0 implies \Ilim as_\alpha=ax_0\Ilim as_\alpha=ax_0. And the proof given in the paper seems to be unnecessary complicated.

We know that f \colon x\mapsto axf \colon x\mapsto ax is a continuous function. (This is true for arbitrary topological vector space.)

So for any convergent net and any ideal we have

s_\alpha \to x_0 \qquad \implies \qquad f(s_\alpha) \to f(x_0),s_\alpha \to x_0 \qquad \implies \qquad f(s_\alpha) \to f(x_0),

i.e.,

as_\alpha \to ax_0 \qquad \implies \qquad as_\alpha \to ax_0.as_\alpha \to ax_0 \qquad \implies \qquad as_\alpha \to ax_0.

Similarly, to see that Theorem 3.6(ii) holds, we can just use continuity of addition as the map V\times V\to VV\times V\to V, and the claim immediately follows. (Basically, I am just saying that we do not need to reprove from scratch facts which are already known, or things which can be easily derived from some well-known facts.)

1Since (based on the comments) this part seemed unclear: If x\in sVx\in sV, this means that x=v_1+\dots v_sx=v_1+\dots v_s for some v_1,\dots,v_2\in Vv_1,\dots,v_2\in V. And we can now rewrite xx as

x=v_1+\dots v_s+\underset{\text{$(t-s)$-times}}{\underbrace{0+\dots+0}}x=v_1+\dots v_s+\underset{\text{$(t-s)$-times}}{\underbrace{0+\dots+0}}

to see that xx also belongs to tVtV.

Thank you. Actually after posting the question this occurred to me : VV and V\times \{0\}V\times \{0\} are homeomorphic and V\times \{0\} \subset V\times VV\times \{0\} \subset V\times V imply that V\subset V\times \{0\}V\subset V\times \{0\}.

– user118494

2 days ago

V\subset V\times\{0\}V\subset V\times\{0\} is certainly not true. The latter is a set of ordered pairs.

– Martin Sleziak

2 days ago

Sorry sorry. I meant V\subset V\times V.V\subset V\times V.

– user118494

2 days ago

That has still the same problem. (And it is not true in general.)

– Martin Sleziak

2 days ago

then how can we say nV\subseteq 2^kVnV\subseteq 2^kV for n\le 2^k\ ?n\le 2^k\ ?

– user118494

2 days ago