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## Park City Mathematics InstituteUndergraduate Summer School 2018

### Introduction to Harmonic Analysis

1. Show that $||\cdot||_{H^1}$ is a norm, by showing it is induced by an inner product.

2. Consider, for a connected domain $\Omega$, the energy form $\mathscr E(u,v) = \int_\Omega \nabla u\cdot \nabla v$.

1. $\mathscr E(u,v)$ is an inner product on $H^1$ modulo constants.

2. $\mathscr E(u,v)$ is an inner product on $H_0^1$.

3. Show the equivalences of the Dirichlet principle.