Introduction

Since our last worksheet, we have been discussing the vector space of square integrable functions, ${\cal L}^{2}(E)$. First, a function $f: E \rightarrow {\bf R} \cup \infty$ is square integrable if

$$\int_{E} \mid f \mid^{2} dm < \infty.$$

Two square integrable functions f and g are considered to be equivalent if they differ at most on a set of measure. Equivalently, f - g = 0 a.e. on E. Then ${\cal L}^{2}(E)$ is the collection of all equivalence classes of square integrable functions.

We defined a metric on ${\cal L}^{2}(E)$ by $d^{2}([f],[g]) = \int_{E} (f - g)^2 dm$. Our first task was to show that ${\cal L}^{2}(E)$ is complete. The first part of this worksheet takes you through this result and the necessary background material. The second part addresses the algebraic questions that arise when we give ${\cal L}^{2}(E)$ the structure of an inner product space.