Question L1 with counting measure

Here is my problem.
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Let Y=L1(m),m is counting measure on N(Natural number),
and let X = {f in Y : InfiniteSum{ n*|f(n)| } is finite },
equipped with the L1 norm.

a.X is proper dense subspace and so X is not complete

b.Define T:X->Y by,Tf(n)=nf(n). Then T is closed but not bounded.

c.Let S=T^{-1}. Then S:Y->X is bounded and surjective but not open.

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a.
As far as I showed X is proper subspace of Y. To show density
I think I have to constuct a sequence of functions, say f_n in X,
which converges to any given function g in Y. Now I'm trying to
build this sequence but it's not easy. Any idea? and if I success to
show there is a converging sequence that implies automatically
X is not complete? Why?
b.
First I tried to use closed graph theorem but I think the function is
not linear so I failed.
c.
How do we know there is an inverse function of T? Is 1/T a specific
inverse function?

Thank you for any help, hint or remark.

.

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