Exercise 2, § 3

For a set S let F = $\mathscr{B}(S)$ be the Banach space of the real-valued bounded functions in S (I, p. 4) and let M be a vector subspace of a normed space E. Show that, for every continuous linear mapping f of M in F, there exists a continuous linear mapping g of E in F, that is an extension of f and such that $\|g\| = \|f\|$.