Day 237: Separable Hilbert Spaces

Learning Objectives

• •**Define** separability for Hilbert spaces
• •**Prove** that a Hilbert space is separable iff it has a countable orthonormal basis
• •**State** the classification theorem for separable Hilbert spaces
• •**Construct** explicit isomorphisms between $$L^2$$ and $$\ell^2$$
• •**Understand** the Riesz representation theorem for Hilbert spaces
• •**Connect** separability to the structure of quantum mechanics

Today's Schedule (7 hours)