• Assume the function space \(X\) is assigned the norm \(\norm{\cdot}\). Show that the definition of \(\norm{\cdot}\) is really a norm, i.e. it should satisfy positive definiteness, scalar multiplication and triangle inequality.

  • \((X,\norm{\cdot})\) is a linear space, i.e. closeness of addition and scalar multiplication should be proved in the sense of finite norm.

  • Prove every Cauchy sequence in \(X\) is convergent.

    • Select a Cauchy sequence \(\left\langle u_n \right\rangle\) in the function space \(X\) and show that it converges to a function \(u\) pointwise. This is natural as long as the codomain of \(\left\langle u_n \right\rangle\) which contain their actual ranges is complete, which can be ensured when the codomain is \(\mathbb{R}\) or \(\mathbb{C}\).

    • Prove \(u \in X\) in the sense of finite norm.

    • Up to now, the convergence of \(u_n\) to \(u\) is still in the pointwise sense. Hence we still need to prove \(u_n\) converges to \(u\) in the norm assigned to \(X\).