Day 234: Hilbert Spaces and L²

Learning Objectives

• •**Define** a Hilbert space as a complete inner product space
• •**Explain** the construction of $$L^2$$ via Lebesgue integration
• •**State** the Riesz-Fischer theorem and understand its significance
• •**Work with** concrete examples in $$L^2[a,b]$$ and $$\ell^2$$
• •**Prove** that closed subspaces of Hilbert spaces are Hilbert spaces
• •**Connect** $$L^2$$ to quantum wave functions and the Born rule

Today's Schedule (7 hours)