The Riesz-Fischer Theorem

Return to the formal definition of $L^{p}$.

Let $(X, \rho )$ denote a generic metric space. Let $\mathscr {C}_{b}(X)$ denote the collection of continuous real-valued bounded functions on $X$. For $f \in \mathscr {C}_{b}(X)$, write $\left\| f\right\| =\sup_{x \in X}\left|f(x)\right|$, the usual supremum norm. Then $\mathscr {C}_{b}(X)$ is a Banach space.

A function $f$ on $\mathbb {R}^{k}$ is said to vanish at infinity iff $f(\boldsymbol {x}) \rightarrow 0$ as $\left\| \boldsymbol {x}\right\| \rightarrow \infty$. Show that the collection of continuous functions on $\mathbb {R}^{k}$ that vanish at infinity is a Banach space relative to the supremum norm given in the previous exercise.

Let $\mathscr {C}_{c}\left(\mathbb {R}^{k}\right)$ denote the collection of continuous functions on $\mathbb {R}^{k}$ with compact support, and again consider the supremum norm. This collection is dense in the collection in the previous exercise, but it fails to be a Banach space.

A sequence $\left\{ f_{n}\right\}$ in $L^{p}$ may converge in $p$ th mean to some $f \in L^{p}$ but at the same time fail to converge pointwise to $f$ at any point in $\Omega$. Therefore, convergence in $L^{p}$ does not in general imply convergence $\mu -$ a. e.