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Let $H$ be a complex Hibert space

Let $S_n$ be a sequence of subsets of $H$ such that for each $n \in \mathbb N$: $S_n$ is compact, $S_n$ is connected, $S_{n+1}\subsetneq S_n$, $\bigcap_{n=1}^\infty S_n=\{v_0\}$, $S_n$ is linear independent

Let, for each $n \in \mathbb N$, be $V_n= \overline{span(S_n)}$ and $T_n:V_n\to\mathbb C$ continuous linear functional such that $T_n(v_0)=0$

I would like to know if $\lim_{n\to\infty}\lVert T_n \rVert = 0$

Thanks.

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    Is such a sequence $S_n$ obtainable? – 2017-01-04
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    yes, H is complete – 2017-01-04
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    Do you have some motivation for this question? Looking at your last questions https://math.stackexchange.com/questions/2078269/sequence-of-continuous-linear-functionals-over-a-sequence-of-hilbert-spaces and https://math.stackexchange.com/questions/2078358/sequence-of-subspace-and-their-continuous-linear-functional they are all ver similar and it looks like you are after something specific and not quite getting the answer you want. Additionally: do you have any argument why such sets $S_n$ should exist? – 2017-01-04
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    hi @LeBtz the reason is that I like to ask myself some problems and then try to solve them. I am not a mathematician nor a graduate but I have a passion for mathematics. – 2017-01-05

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