I feel like I must be missing a trick - I'm self studying functional analysis and have come across Schauder bases, and I'm looking at different classifications e.g. shrinking, boundedly complete, unconditional bases.
A basis $\{e_i; e_i^*\}$ for a Banach space $X$ is shrinking if $\overline{\text{span}}\{e_i^*\} = X^*$, or in some books, if $\{e_i^*; e_i\}$ is a basis for $X^*$.
I understand this definition, and proofs of the first results about shrinking bases e.g. $\{e_i; e_i^*\}$ is shrinking if and only if $$ \lim_{n \to \infty} \left\lVert x_{\overline{\text{span}}\{e_i\}_{i \geq n}}^* \right\rVert = 0 $$ for every $x^* \in X^*$. But in all the books I've looked at, the author will follow up the definition saying it is immediate, obvious, clear that the standard basis for $c_0$ is shrinking, and for $\ell_p$ whenever $p>1$, but not for $\ell_1$. The last statement is clear to me, since $\ell_1^* \cong \ell_{\infty}$ is not separable so it can't have a basis, but I have no idea how one would show the others? Should I try and construct an isometric isomorphism between $\overline{\text{span}}\{e_i\}$ and $\ell_1$ for the case of $c_0$? I don't see how to do that without being able to describe a general element in $\overline{\text{span}}\{e_i\}$. Or should I use something like the equivalent condition I stated?
A standard result is that a Banach space with a Schauder basis is reflexive iff the basis is shrinking and boundedly complete. The spaces $\ell_{p}$ for $1<p<\infty$ are reflexive so your result is immediate if you know this (James') theorem. For the $c_{0}$ case, note that $c_{0}^{*}\cong\ell_{1}$ and apply the definition of a shrinking basis.