Let \(\ell ^\infty \) be the normed space consisting of all complex valued, bounded sequences \[ \mathbf a=(a_1,a_2,\cdots a_n\cdots ) \] with the norm \(\lVert \mathbf a\rVert =\sup \{|a_n|:n\in \mathbb N\}\). Let \(W\subset \ell ^\infty \) be the subspace consisting of all \(\mathbf a\in \ell ^\infty \) such that \(\{n\in \mathbb N:a_n\ne 0\}\) is a finite set.
Show that there exists a nontrivial bounded linear functional on \(\ell ^\infty \) which vanishes on \(W\).