Recasting the Proof of Parseval's Identity

Abstract: We generalize aspects of Fourier Analysis from intervals on $\mathbb{R}$ to bounded and measurable subsets of $\mathbb{R}<sup>n$. In doing so, we obtain a few interesting results. The first is a new proof of the famous Integral Cauchy-Schwarz Inequality. The second is a restatement of Parseval's Identity that doubles as a representation of integrating bounded and measurable functions over bounded and measurable subsets of $\mathbb{R}<sup>n$. Finally, we apply these first two results to develop some sufficient criteria for additional integral inequalities that are elementary in nature.