The Classical Spaces

In this chapter, the properties of the main normed spaces are explored: \(\ell ^\infty \) is complete but not separable; it contains the separable closed subspace \(c_0\) . The space \(\ell ^1\) is complete and separable, and is the dual space of \(c_0\) . These results are generalized for \(\ell ^p\) , using Minkowski’s and Hölder’s inequalities, with \(\ell ^2\) being its own dual. After a review of measures, measurable functions, and integrability, the spaces \(L^1(\mathbb {R})\) and \(L^\infty (\mathbb {R})\) are shown to be Banach spaces. Various theorems about the approximation of functions by polynomials and the “Approximation of the Identity” lemma are made, with an application to Fourier series and its properties.