I posted my question at MS but unfortunately it is still without a response, so let me ask it here.


We can think about a bounded operator $T\colon c\_0\to c\_0$ as a double-infinite matrix $[T\_{mn}]\_{m,n\geq 1}$ which acts on a sequence $a=[a\_1, a\_2, a\_3, \ldots ]\in c\_0$ in the same way as usual (finite) matrices act on vectors ($n$-tuples of scalars), i.e.


$$ Ta= [T\_{mn}][a\_n] = \left[ \sum\_{n=1}^\infty T\_{mn}a\_n\right] $$


Suppose $a=[a\_1, a\_2, a\_3, \ldots ]\in \ell^\infty = (c\_0)^{\*\*}$. Does the following formula still hold:


$$ T^{\*\*}a= [T\_{mn}][a\_n] = \left[ \sum\_{n=1}^\infty T\_{mn}a\_n\right] $$


EDIT: Second question deleted, since I wanted to ask, in fact, about something else.



Second conjugate operators to operators on $c_0$
Yes. The reason is that the unit vector basis for $c\_0$ is a shrinking basis, which means that the biorthogonal functionals to the basis are a Schauder basis for $c\_0^\* = \ell\_1$. This implies that the unit vector basis for $c\_0$ is a weak$^\*$ Schauder basis for $c\_0^{\*\*} = \ell\_\infty$, which means that the expansion of a vector in $\ell\_\infty$ in terms of the unit vector basis must converge in the weak$^\*$ topology. This is enough, since the adjoint of an operator on $\ell\_1$ is weak$^\*$ continuous on $\ell\_\infty$. 


I'll take this opportunity to say something more about a space with shrinking bases. R. C. James, who introduced the concept, realized that if $(e\_n)$ is a shrinking basis for a space $X$, then $X^{\*\*}$ can be represented as the space of sequences $(a\_n)$ s.t. the partial sums of the series $\sum a\_n e\_n$ are uniformly bounded. Once he realized this, he very quickly constructed the space we now call J, which is isomorphic to its second dual but is non reflexive; in fact, $\dim J^{\*\*}/J =1$.


You can read about the basics of basis theory in many books; Albiac-Kalton is a good choice.


