You're correct about the definitions of the spaces (and yes we only consider bounded sequences). The notation is a bit sloppy, but viewing
c0⊂ℓ∞, a projection in this context is a bounded linear map
P:ℓ∞→ℓ∞ such that
P2=P and
P(ℓ∞)=c0. Also in the context of functional analysis we require all maps to be continuous, so it's much harder for spaces to be isomorphic unlike in linear algebra (we need a linear homeomorphism). In the same spirit, if
X is a normed space then its dual
X∗ is the set of continuous linear functionals
P:X→R or
C.
Proving this claim takes a bit more work, but the idea is to apply the result to
S=N and show that if
f∈(ℓ∞)∗ then the set
{i∈I:f(χAi)=0} is countable (
χAi is the indicator function/sequence of each
Ai). If such a projection
P exists, then
c0=Ker(id−P)=⋂n∈Nen∗∘(id−P), where each
en∗∈(ℓ∞)∗ is the nth coordinate functional. But this is an intersection of linear functionals, so by cardinality considerations it contains some
χAi, which is a contradiction as each
χAi∈c0.
The last claim follows from a general fact that if
X is a normed space, there exists a bounded projection
P:X∗∗∗→X∗. The idea is to take the canonical embedding
i:X→X∗∗ and show its adjoint is a projection making appropriate identifications. We can apply this result here since
c0∗∗≅ℓ∞, making sure everything still works up to isomorphism.