  We generalize aspects of Fourier Analysis from intervals on $\mathbb{R}$ to
bounded and measurable subsets of $\mathbb{R}^n$. 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 $\mathbb{R}^n$. Finally, we
apply these first two results to develop some sufficient criteria for
additional integral inequalities that are elementary in nature.
