I was working on the following problem:
Let $L^\infty(\mathbf{R})$ denote the essentially bounded functions and let $BC(\mathbf{R})$ denote the bounded continuous functions. Show that there is some $\ell \in (\mathbf{L}^\infty(\mathbf{R})^\ast$ such that $\ell(f) = f(0)$ for each $f \in BC(\mathbf{R})$.
There are two things I could see to do here. The first is apply BLT theorem. Since $\ell(f) = f(0)$ is a bounded linear map on $BC(\mathbf{R})$ this map extends to the completion of $BC(\mathbf{R})$. So it would suffice to show that any $L^\infty(\mathbf{R})$ function is the limit of $BC(\mathbf{R})$ functions. The other thing to do is to try applyinh Hahn-Banach theorem: which seems more straightforward. $BC(\mathbf{R})$ is a linear subspace of $L^\infty(\mathbf{R})$ thus, defining a map $f \mapsto f(0)$ on $BC(\mathbf{R})$ which is clearly a bounded linear map extends to $\ell$ with the same norm (1), on $L^\infty(\mathbf{R})$. Which is the correct way to solve the problem?